Misplaced Pages

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125-419: There are two main descriptions of motion: dynamics and kinematics . Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations ), and sometimes to the solutions to those equations. However, kinematics

250-416: A B x , a B y , a B z ) {\displaystyle \mathbf {a} _{B}=\left(a_{B_{x}},a_{B_{y}},a_{B_{z}}\right)} then the acceleration of point C relative to point B is the difference between their components: a C / B = a C − a B = (

375-460: A C x − a B x , a C y − a B y , a C z − a B z ) {\displaystyle \mathbf {a} _{C/B}=\mathbf {a} _{C}-\mathbf {a} _{B}=\left(a_{C_{x}}-a_{B_{x}},a_{C_{y}}-a_{B_{y}},a_{C_{z}}-a_{B_{z}}\right)} Alternatively, this same result could be obtained by computing

500-457: A t 2 2 + v 0 t + r 0 , [ 2 ] {\displaystyle {\begin{aligned}\mathbf {v} &=\int \mathbf {a} dt=\mathbf {a} t+\mathbf {v} _{0}\,,&[1]\\\mathbf {r} &=\int (\mathbf {a} t+\mathbf {v} _{0})dt={\frac {\mathbf {a} t^{2}}{2}}+\mathbf {v} _{0}t+\mathbf {r} _{0}\,,&[2]\\\end{aligned}}} in magnitudes, v =

625-443: A | = | v ˙ | = d v d t . {\displaystyle |\mathbf {a} |=|{\dot {\mathbf {v} }}|={\frac {{\text{d}}v}{{\text{d}}t}}.} A relative position vector is a vector that defines the position of one point relative to another. It is the difference in position of the two points. The position of one point A relative to another point B

750-541: A τ ) d τ = r 0 + v 0 t + 1 2 a t 2 . {\displaystyle \mathbf {r} (t)=\mathbf {r} _{0}+\int _{0}^{t}\mathbf {v} (\tau )\,{\text{d}}\tau =\mathbf {r} _{0}+\int _{0}^{t}\left(\mathbf {v} _{0}+\mathbf {a} \tau \right){\text{d}}\tau =\mathbf {r} _{0}+\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}.} Additional relations between displacement, velocity, acceleration, and time can be derived. Since

875-577: A ( r − r 0 ) [ 4 ] r = r 0 + v t − 1 2 a t 2 [ 5 ] {\displaystyle {\begin{aligned}v&=at+v_{0}&[1]\\r&=r_{0}+v_{0}t+{\tfrac {1}{2}}{a}t^{2}&[2]\\r&=r_{0}+{\tfrac {1}{2}}\left(v+v_{0}\right)t&[3]\\v^{2}&=v_{0}^{2}+2a\left(r-r_{0}\right)&[4]\\r&=r_{0}+vt-{\tfrac {1}{2}}{a}t^{2}&[5]\\\end{aligned}}} where: Equations [1] and [2] are from integrating

1000-409: A r = − v θ , a θ = a , {\displaystyle a_{r}=-v\theta ,\quad a_{\theta }=a,} are called, respectively, the radial and tangential components of acceleration. Momentum In Newtonian mechanics , momentum ( pl. : momenta or momentums ; more specifically linear momentum or translational momentum )

1125-439: A x x ^ + a y y ^ + a z z ^ . {\displaystyle \mathbf {a} =\lim _{(\Delta t)^{2}\to 0}{\frac {\Delta \mathbf {r} }{(\Delta t)^{2}}}={\frac {{\text{d}}^{2}\mathbf {r} }{{\text{d}}t^{2}}}=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}.} Thus, acceleration

1250-616: A y y ^ + a z z ^ . {\displaystyle \mathbf {a} =\lim _{\Delta t\to 0}{\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {{\text{d}}\mathbf {v} }{{\text{d}}t}}=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}.} Alternatively, a = lim ( Δ t ) 2 → 0 Δ r ( Δ t ) 2 = d 2 r d t 2 =

1375-638: A ¯ x x ^ + a ¯ y y ^ + a ¯ z z ^ {\displaystyle \mathbf {\bar {a}} ={\frac {\Delta \mathbf {\bar {v}} }{\Delta t}}={\frac {\Delta {\bar {v}}_{x}}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta {\bar {v}}_{y}}{\Delta t}}{\hat {\mathbf {y} }}+{\frac {\Delta {\bar {v}}_{z}}{\Delta t}}{\hat {\mathbf {z} }}={\bar {a}}_{x}{\hat {\mathbf {x} }}+{\bar {a}}_{y}{\hat {\mathbf {y} }}+{\bar {a}}_{z}{\hat {\mathbf {z} }}\,} where Δ v

SECTION 10

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1500-564: A + v ω ) θ ^ + a z z ^ . {\displaystyle \mathbf {a} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(v{\hat {\mathbf {r} }}+v{\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}\right)=(a-v\theta ){\hat {\mathbf {r} }}+(a+v\omega ){\hat {\mathbf {\theta } }}+a_{z}{\hat {\mathbf {z} }}.} The term − v θ r ^ {\displaystyle -v\theta {\hat {\mathbf {r} }}} acts toward

1625-402: A n t . {\displaystyle m_{A}v_{A}+m_{B}v_{B}+m_{C}v_{C}+...=constant.} This conservation law applies to all interactions, including collisions (both elastic and inelastic ) and separations caused by explosive forces. It can also be generalized to situations where Newton's laws do not hold, for example in the theory of relativity and in electrodynamics . Momentum

1750-410: A t + v 0 [ 1 ] r = r 0 + v 0 t + 1 2 a t 2 [ 2 ] r = r 0 + 1 2 ( v + v 0 ) t [ 3 ] v 2 = v 0 2 + 2

1875-456: A t + v 0 , [ 1 ] r = a t 2 2 + v 0 t + r 0 . [ 2 ] {\displaystyle {\begin{aligned}v&=at+v_{0}\,,&[1]\\r&={\frac {{a}t^{2}}{2}}+v_{0}t+r_{0}\,.&[2]\\\end{aligned}}} Equation [3] involves the average velocity ⁠ v + v 0 / 2 ⁠ . Intuitively,

2000-486: A t t = 1 2 a t 2 = a t 2 2 {\textstyle A={\frac {1}{2}}BH={\frac {1}{2}}att={\frac {1}{2}}at^{2}={\frac {at^{2}}{2}}} . Adding v 0 t {\displaystyle v_{0}t} and a t 2 2 {\textstyle {\frac {at^{2}}{2}}} results in the equation Δ r {\displaystyle \Delta r} results in

2125-463: A = a ( t ) have the general, coordinate-independent definitions; v = d r d t , a = d v d t = d 2 r d t 2 {\displaystyle \mathbf {v} ={\frac {d\mathbf {r} }{dt}}\,,\quad \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}} Notice that velocity always points in

2250-418: A Galilean transformation . If a particle is moving at speed ⁠ d x / d t ⁠ = v in the first frame of reference, in the second, it is moving at speed v ′ = d x ′ d t = v − u . {\displaystyle v'={\frac {{\text{d}}x'}{{\text{d}}t}}=v-u\,.} Since u does not change,

2375-512: A momentum density can be defined as momentum per volume (a volume-specific quantity ). A continuum version of the conservation of momentum leads to equations such as the Navier–Stokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids. Momentum is a vector quantity : it has both magnitude and direction. Since momentum has a direction, it can be used to predict

2500-752: A 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg⋅m/s due north measured with reference to the ground. The momentum of a system of particles is the vector sum of their momenta. If two particles have respective masses m 1 and m 2 , and velocities v 1 and v 2 , the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2 . {\displaystyle {\begin{aligned}p&=p_{1}+p_{2}\\&=m_{1}v_{1}+m_{2}v_{2}\,.\end{aligned}}} The momenta of more than two particles can be added more generally with

2625-402: A collision. For example, suppose there are two bodies of equal mass m , one stationary and one approaching the other at a speed v (as in the figure). The center of mass is moving at speed ⁠ v / 2 ⁠ and both bodies are moving towards it at speed ⁠ v / 2 ⁠ . Because of the symmetry, after the collision both must be moving away from the center of mass at

SECTION 20

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2750-628: A convenient form. Recall that the trajectory of a particle P is defined by its coordinate vector r measured in a fixed reference frame F . As the particle moves, its coordinate vector r ( t ) traces its trajectory, which is a curve in space, given by: r ( t ) = x ( t ) x ^ + y ( t ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=x(t){\hat {\mathbf {x} }}+y(t){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where x̂ , ŷ , and ẑ are

2875-437: A formula relating time, velocity and distance. De Soto's comments are remarkably correct regarding the definitions of acceleration (acceleration was a rate of change of motion (velocity) in time) and the observation that acceleration would be negative during ascent. Discourses such as these spread throughout Europe, shaping the work of Galileo Galilei and others, and helped in laying the foundation of kinematics. Galileo deduced

3000-531: A mechanism for a desired range of motion. In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system or mechanism. The term kinematic is the English version of A.M. Ampère 's cinématique , which he constructed from the Greek κίνημα kinema ("movement, motion"), itself derived from κινεῖν kinein ("to move"). Kinematic and cinématique are related to

3125-627: A new body of knowledge, now called physics. At Oxford, Merton College sheltered a group of scholars devoted to natural science, mainly physics, astronomy and mathematics, who were of similar stature to the intellectuals at the University of Paris. Thomas Bradwardine extended Aristotelian quantities such as distance and velocity, and assigned intensity and extension to them. Bradwardine suggested an exponential law involving force, resistance, distance, velocity and time. Nicholas Oresme further extended Bradwardine's arguments. The Merton school proved that

3250-954: A particle trajectory on a circular cylinder occurs when there is no movement along the z axis: r ( t ) = r r ^ + z z ^ , {\displaystyle \mathbf {r} (t)=r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }},} where r and z 0 are constants. In this case, the velocity v P is given by: v P = d d t ( r r ^ + z z ^ ) = r ω θ ^ = v θ ^ , {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=r\omega {\hat {\mathbf {\theta } }}=v{\hat {\mathbf {\theta } }},} where ω {\displaystyle \omega }

3375-632: A particle's velocity is the time rate of change of its position. Furthermore, this velocity is tangent to the particle's trajectory at every position along its path. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants. The speed of an object is the magnitude of its velocity. It is a scalar quantity: v = | v | = d s d t , {\displaystyle v=|\mathbf {v} |={\frac {{\text{d}}s}{{\text{d}}t}},} where s {\displaystyle s}

3500-819: A reference frame. The position vector of a particle is a vector drawn from the origin of the reference frame to the particle. It expresses both the distance of the point from the origin and its direction from the origin. In three dimensions, the position vector r {\displaystyle {\bf {r}}} can be expressed as r = ( x , y , z ) = x x ^ + y y ^ + z z ^ , {\displaystyle \mathbf {r} =(x,y,z)=x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }},} where x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} are

3625-440: A result of greater elevation. Only Domingo de Soto , a Spanish theologian, in his commentary on Aristotle 's Physics published in 1545, after defining "uniform difform" motion (which is uniformly accelerated motion) – the word velocity was not used – as proportional to time, declared correctly that this kind of motion was identifiable with freely falling bodies and projectiles, without his proving these propositions or suggesting

3750-551: A time interval is defined as the ratio. a ¯ = Δ v ¯ Δ t = Δ v ¯ x Δ t x ^ + Δ v ¯ y Δ t y ^ + Δ v ¯ z Δ t z ^ =

3875-401: Is a unit vector in the direction of the axis of rotation, and θ is the angle the object turns through about the axis. The following relation holds for a point-like particle, orbiting about some axis with angular velocity ω : v = ω × r {\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} } where r

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4000-413: Is a good example of an almost totally elastic collision, due to their high rigidity , but when bodies come in contact there is always some dissipation . A head-on elastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are v A1 and v B1 before the collision and v A2 and v B2 after,

4125-599: Is a measurable quantity, and the measurement depends on the frame of reference . For example: if an aircraft of mass 1000 kg is flying through the air at a speed of 50 m/s its momentum can be calculated to be 50,000 kg.m/s. If the aircraft is flying into a headwind of 5 m/s its speed relative to the surface of the Earth is only 45 m/s and its momentum can be calculated to be 45,000 kg.m/s. Both calculations are equally correct. In both frames of reference, any change in momentum will be found to be consistent with

4250-445: Is a subfield of physics and mathematics , developed in classical mechanics , that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of both applied and pure mathematics since it can be studied without considering

4375-466: Is an inelastic collision . An elastic collision is one in which no kinetic energy is transformed into heat or some other form of energy. Perfectly elastic collisions can occur when the objects do not touch each other, as for example in atomic or nuclear scattering where electric repulsion keeps the objects apart. A slingshot maneuver of a satellite around a planet can also be viewed as a perfectly elastic collision. A collision between two pool balls

4500-401: Is defined as the coordinate vector from the origin of a coordinate frame to the particle. For example, consider a tower 50 m south from your home, where the coordinate frame is centered at your home, such that east is in the direction of the x -axis and north is in the direction of the y -axis, then the coordinate vector to the base of the tower is r = (0 m, −50 m, 0 m). If

4625-605: Is equal to the instantaneous force F acting on it, F = d p d t . {\displaystyle F={\frac {{\text{d}}p}{{\text{d}}t}}.} If the net force experienced by a particle changes as a function of time, F ( t ) , the change in momentum (or impulse J ) between times t 1 and t 2 is Δ p = J = ∫ t 1 t 2 F ( t ) d t . {\displaystyle \Delta p=J=\int _{t_{1}}^{t_{2}}F(t)\,{\text{d}}t\,.} Impulse

4750-407: Is known as Euler's first law . If the net force F applied to a particle is constant, and is applied for a time interval Δ t , the momentum of the particle changes by an amount Δ p = F Δ t . {\displaystyle \Delta p=F\Delta t\,.} In differential form, this is Newton's second law ; the rate of change of the momentum of a particle

4875-468: Is measured in the derived units of the newton second (1 N⋅s = 1 kg⋅m/s) or dyne second (1 dyne⋅s = 1 g⋅cm/s) Under the assumption of constant mass m , it is equivalent to write F = d ( m v ) d t = m d v d t = m a , {\displaystyle F={\frac {{\text{d}}(mv)}{{\text{d}}t}}=m{\frac {{\text{d}}v}{{\text{d}}t}}=ma,} hence

5000-561: Is more likely to be exactly solvable. In general, the equation will be non-linear , and cannot be solved exactly so a variety of approximations must be used. The solutions to nonlinear equations may show chaotic behavior depending on how sensitive the system is to the initial conditions. Kinematics, dynamics and the mathematical models of the universe developed incrementally over three millennia, thanks to many thinkers, only some of whose names we know. In antiquity, priests , astrologers and astronomers predicted solar and lunar eclipses ,

5125-560: Is not affected by external forces, its total momentum does not change. Momentum is also conserved in special relativity (with a modified formula) and, in a modified form, in electrodynamics , quantum mechanics , quantum field theory , and general relativity . It is an expression of one of the fundamental symmetries of space and time: translational symmetry . Advanced formulations of classical mechanics, Lagrangian and Hamiltonian mechanics , allow one to choose coordinate systems that incorporate symmetries and constraints. In these systems

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5250-466: Is numerically equivalent to 3 newtons. In a closed system (one that does not exchange any matter with its surroundings and is not acted on by external forces) the total momentum remains constant. This fact, known as the law of conservation of momentum , is implied by Newton's laws of motion . Suppose, for example, that two particles interact. As explained by the third law, the forces between them are equal in magnitude but opposite in direction. If

5375-466: Is of prime importance. He measured momentum by the product of velocity and weight; mass is a later concept, developed by Huygens and Newton. In the swinging of a simple pendulum, Galileo says in Discourses that "every momentum acquired in the descent along an arc is equal to that which causes the same moving body to ascend through the same arc." His analysis on projectiles indicates that Galileo had grasped

5500-580: Is simpler. It concerns only variables derived from the positions of objects and time. In circumstances of constant acceleration, these simpler equations of motion are usually referred to as the SUVAT equations , arising from the definitions of kinematic quantities : displacement ( s ), initial velocity ( u ), final velocity ( v ), acceleration ( a ), and time ( t ). A differential equation of motion, usually identified as some physical law (for example, F = ma) and applying definitions of physical quantities ,

5625-602: Is simply the difference between their positions which is the difference between the components of their position vectors. If point A has position components r A = ( x A , y A , z A ) {\displaystyle \mathbf {r} _{A}=\left(x_{A},y_{A},z_{A}\right)} and point B has position components r B = ( x B , y B , z B ) {\displaystyle \mathbf {r} _{B}=\left(x_{B},y_{B},z_{B}\right)} then

5750-882: Is simply the difference between their velocities v A / B = v A − v B {\displaystyle \mathbf {v} _{A/B}=\mathbf {v} _{A}-\mathbf {v} _{B}} which is the difference between the components of their velocities. If point A has velocity components v A = ( v A x , v A y , v A z ) {\displaystyle \mathbf {v} _{A}=\left(v_{A_{x}},v_{A_{y}},v_{A_{z}}\right)} and point B has velocity components v B = ( v B x , v B y , v B z ) {\displaystyle \mathbf {v} _{B}=\left(v_{B_{x}},v_{B_{y}},v_{B_{z}}\right)} then

5875-594: Is the angular velocity of the unit vector θ around the z axis of the cylinder. The acceleration a P of the particle P is now given by: a P = d ( v θ ^ ) d t = a θ ^ − v θ r ^ . {\displaystyle \mathbf {a} _{P}={\frac {{\text{d}}(v{\hat {\mathbf {\theta } }})}{{\text{d}}t}}=a{\hat {\mathbf {\theta } }}-v\theta {\hat {\mathbf {r} }}.} The components

6000-422: Is the center of mass frame – one that is moving with the center of mass. In this frame, the total momentum is zero. If two particles, each of known momentum, collide and coalesce, the law of conservation of momentum can be used to determine the momentum of the coalesced body. If the outcome of the collision is that the two particles separate, the law is not sufficient to determine the momentum of each particle. If

6125-490: Is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If m is an object's mass and v is its velocity (also a vector quantity), then the object's momentum p (from Latin pellere "push, drive") is: p = m v . {\displaystyle \mathbf {p} =m\mathbf {v} .} In the International System of Units (SI),

6250-417: Is the arc-length measured along the trajectory of the particle. This arc-length must always increase as the particle moves. Hence, d s / d t {\displaystyle {\text{d}}s/{\text{d}}t} is non-negative, which implies that speed is also non-negative. The velocity vector can change in magnitude and in direction or both at once. Hence, the acceleration accounts for both

6375-411: Is the area under a velocity–time graph. We can take Δ r {\displaystyle \Delta r} by adding the top area and the bottom area. The bottom area is a rectangle, and the area of a rectangle is the A ⋅ B {\displaystyle A\cdot B} where A {\displaystyle A} is the width and B {\displaystyle B}

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6500-441: Is the average velocity and Δ t is the time interval. The acceleration of the particle is the limit of the average acceleration as the time interval approaches zero, which is the time derivative, a = lim Δ t → 0 Δ v Δ t = d v d t = a x x ^ +

6625-948: Is the displacement vector during the time interval Δ t {\displaystyle \Delta t} . In the limit that the time interval Δ t {\displaystyle \Delta t} approaches zero, the average velocity approaches the instantaneous velocity, defined as the time derivative of the position vector, v = lim Δ t → 0 Δ r Δ t = d r d t = v x x ^ + v y y ^ + v z z ^ . {\displaystyle \mathbf {v} =\lim _{\Delta t\to 0}{\frac {\Delta \mathbf {r} }{\Delta t}}={\frac {{\text{d}}\mathbf {r} }{{\text{d}}t}}=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}.} Thus,

6750-406: Is the first derivative of the velocity vector and the second derivative of the position vector of that particle. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants. The magnitude of the acceleration of an object is the magnitude | a | of its acceleration vector. It is a scalar quantity: |

6875-423: Is the height. In this case A = t {\displaystyle A=t} and B = v 0 {\displaystyle B=v_{0}} (the A {\displaystyle A} here is different from the acceleration a {\displaystyle a} ). This means that the bottom area is t v 0 {\displaystyle tv_{0}} . Now let's find

7000-408: Is the position vector of the particle (radial from the rotation axis) and v the tangential velocity of the particle. For a rotating continuum rigid body , these relations hold for each point in the rigid body. The differential equation of motion for a particle of constant or uniform acceleration in a straight line is simple: the acceleration is constant, so the second derivative of the position of

7125-409: Is the product of the units of mass and velocity. In SI units , if the mass is in kilograms and the velocity is in meters per second then the momentum is in kilogram meters per second (kg⋅m/s). In cgs units , if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeters per second (g⋅cm/s). Being a vector, momentum has magnitude and direction. For example,

7250-400: Is time, and each overdot denotes one time derivative . The initial conditions are given by the constant values at t = 0 , r ( 0 ) , r ˙ ( 0 ) . {\displaystyle \mathbf {r} (0)\,,\quad \mathbf {\dot {r}} (0)\,.} The solution r ( t ) to the equation of motion, with specified initial values, describes

7375-455: Is unchanged. Forces such as Newtonian gravity, which depend only on the scalar distance between objects, satisfy this criterion. This independence of reference frame is called Newtonian relativity or Galilean invariance . A change of reference frame, can, often, simplify calculations of motion. For example, in a collision of two particles, a reference frame can be chosen, where, one particle begins at rest. Another, commonly used reference frame,

7500-412: Is used in astrophysics to describe the motion of celestial bodies and collections of such bodies. In mechanical engineering , robotics , and biomechanics , kinematics is used to describe the motion of systems composed of joined parts (multi-link systems) such as an engine , a robotic arm or the human skeleton . Geometric transformations, also called rigid transformations , are used to describe

7625-416: Is used to set up an equation for the problem. Solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a family of solutions. A particular solution can be obtained by setting the initial values , which fixes the values of the constants. To state this formally, in general an equation of motion M is a function of the position r of

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7750-597: The Cartesian coordinates and x ^ {\displaystyle {\hat {\mathbf {x} }}} , y ^ {\displaystyle {\hat {\mathbf {y} }}} and z ^ {\displaystyle {\hat {\mathbf {z} }}} are the unit vectors along the x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} coordinate axes, respectively. The magnitude of

7875-459: The Franck–Hertz experiment ); and particle accelerators in which the kinetic energy is converted into mass in the form of new particles. In a perfectly inelastic collision (such as a bug hitting a windshield), both bodies have the same motion afterwards. A head-on inelastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If

8000-473: The dot product , which is appropriate as the products are scalars rather than vectors. 2 ( r − r 0 ) ⋅ a = | v | 2 − | v 0 | 2 . {\displaystyle 2\left(\mathbf {r} -\mathbf {r} _{0}\right)\cdot \mathbf {a} =|\mathbf {v} |^{2}-|\mathbf {v} _{0}|^{2}.} The dot product can be replaced by

8125-405: The unit of measurement of momentum is the kilogram metre per second (kg⋅m/s), which is dimensionally equivalent to the newton-second . Newton's second law of motion states that the rate of change of a body's momentum is equal to the net force acting on it. Momentum depends on the frame of reference , but in any inertial frame it is a conserved quantity, meaning that if a closed system

8250-842: The unit vectors along the x , y and z axes of the reference frame F , respectively. Consider a particle P that moves only on the surface of a circular cylinder r ( t ) = constant, it is possible to align the z axis of the fixed frame F with the axis of the cylinder. Then, the angle θ around this axis in the x – y plane can be used to define the trajectory as, r ( t ) = r cos ⁡ ( θ ( t ) ) x ^ + r sin ⁡ ( θ ( t ) ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=r\cos(\theta (t)){\hat {\mathbf {x} }}+r\sin(\theta (t)){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where

8375-457: The wavefunction , which describes how a quantum state behaves analogously using the space and time coordinates of the particles. There are analogs of equations of motion in other areas of physics, for collections of physical phenomena that can be considered waves, fluids, or fields. From the instantaneous position r = r ( t ) , instantaneous meaning at an instant value of time t , the instantaneous velocity v = v ( t ) and acceleration

8500-496: The French word cinéma, but neither are directly derived from it. However, they do share a root word in common, as cinéma came from the shortened form of cinématographe, "motion picture projector and camera", once again from the Greek word for movement and from the Greek γρᾰ́φω grapho ("to write"). Particle kinematics is the study of the trajectory of particles. The position of a particle

8625-424: The acceleration a P , which is the time derivative of the velocity v P , is given by: a P = d d t ( v r ^ + v θ ^ + v z z ^ ) = ( a − v θ ) r ^ + (

8750-729: The acceleration is constant, a = Δ v Δ t = v − v 0 t {\displaystyle \mathbf {a} ={\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {\mathbf {v} -\mathbf {v} _{0}}{t}}} can be substituted into the above equation to give: r ( t ) = r 0 + ( v + v 0 2 ) t . {\displaystyle \mathbf {r} (t)=\mathbf {r} _{0}+\left({\frac {\mathbf {v} +\mathbf {v} _{0}}{2}}\right)t.} A relationship between velocity, position and acceleration without explicit time dependence can be had by solving

8875-739: The average acceleration for time and substituting and simplifying t = v − v 0 a {\displaystyle t={\frac {\mathbf {v} -\mathbf {v} _{0}}{\mathbf {a} }}} ( r − r 0 ) ⋅ a = ( v − v 0 ) ⋅ v + v 0 2   , {\displaystyle \left(\mathbf {r} -\mathbf {r} _{0}\right)\cdot \mathbf {a} =\left(\mathbf {v} -\mathbf {v} _{0}\right)\cdot {\frac {\mathbf {v} +\mathbf {v} _{0}}{2}}\ ,} where ⋅ {\displaystyle \cdot } denotes

9000-635: The case of acceleration always in the direction of the motion and the direction of motion should be in positive or negative, the angle between the vectors ( α ) is 0, so cos ⁡ 0 = 1 {\displaystyle \cos 0=1} , and | v | 2 = | v 0 | 2 + 2 | a | | r − r 0 | . {\displaystyle |\mathbf {v} |^{2}=|\mathbf {v} _{0}|^{2}+2\left|\mathbf {a} \right|\left|\mathbf {r} -\mathbf {r} _{0}\right|.} This can be simplified using

9125-455: The center of curvature of the path at that point on the path, is commonly called the centripetal acceleration . The term v ω θ ^ {\displaystyle v\omega {\hat {\mathbf {\theta } }}} is called the Coriolis acceleration . If the trajectory of the particle is constrained to lie on a cylinder, then the radius r is constant and

9250-412: The components of their accelerations. If point C has acceleration components a C = ( a C x , a C y , a C z ) {\displaystyle \mathbf {a} _{C}=\left(a_{C_{x}},a_{C_{y}},a_{C_{z}}\right)} and point B has acceleration components a B = (

9375-496: The conserved quantity is generalized momentum , and in general this is different from the kinetic momentum defined above. The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a wave function . The momentum and position operators are related by the Heisenberg uncertainty principle . In continuous systems such as electromagnetic fields , fluid dynamics and deformable bodies ,

9500-2770: The constant distance from the center is denoted as r , and θ ( t ) is a function of time. The cylindrical coordinates for r ( t ) can be simplified by introducing the radial and tangential unit vectors, r ^ = cos ⁡ ( θ ( t ) ) x ^ + sin ⁡ ( θ ( t ) ) y ^ , θ ^ = − sin ⁡ ( θ ( t ) ) x ^ + cos ⁡ ( θ ( t ) ) y ^ . {\displaystyle {\hat {\mathbf {r} }}=\cos(\theta (t)){\hat {\mathbf {x} }}+\sin(\theta (t)){\hat {\mathbf {y} }},\quad {\hat {\mathbf {\theta } }}=-\sin(\theta (t)){\hat {\mathbf {x} }}+\cos(\theta (t)){\hat {\mathbf {y} }}.} and their time derivatives from elementary calculus: d r ^ d t = ω θ ^ . {\displaystyle {\frac {{\text{d}}{\hat {\mathbf {r} }}}{{\text{d}}t}}=\omega {\hat {\mathbf {\theta } }}.} d 2 r ^ d t 2 = d ( ω θ ^ ) d t = α θ ^ − ω r ^ . {\displaystyle {\frac {{\text{d}}^{2}{\hat {\mathbf {r} }}}{{\text{d}}t^{2}}}={\frac {{\text{d}}(\omega {\hat {\mathbf {\theta } }})}{{\text{d}}t}}=\alpha {\hat {\mathbf {\theta } }}-\omega {\hat {\mathbf {r} }}.} d θ ^ d t = − θ r ^ . {\displaystyle {\frac {{\text{d}}{\hat {\mathbf {\theta } }}}{{\text{d}}t}}=-\theta {\hat {\mathbf {r} }}.} d 2 θ ^ d t 2 = d ( − θ r ^ ) d t = − α r ^ − ω 2 θ ^ . {\displaystyle {\frac {{\text{d}}^{2}{\hat {\mathbf {\theta } }}}{{\text{d}}t^{2}}}={\frac {{\text{d}}(-\theta {\hat {\mathbf {r} }})}{{\text{d}}t}}=-\alpha {\hat {\mathbf {r} }}-\omega ^{2}{\hat {\mathbf {\theta } }}.} Using this notation, r ( t ) takes

9625-590: The constant tangential acceleration is applied along that path , so v 2 = v 0 2 + 2 a Δ r . {\displaystyle v^{2}=v_{0}^{2}+2a\Delta r.} This reduces the parametric equations of motion of the particle to a Cartesian relationship of speed versus position. This relation is useful when time is unknown. We also know that Δ r = ∫ v d t {\textstyle \Delta r=\int v\,{\text{d}}t} or Δ r {\displaystyle \Delta r}

9750-574: The cosine of the angle α between the vectors (see Geometric interpretation of the dot product for more details) and the vectors by their magnitudes, in which case: 2 | r − r 0 | | a | cos ⁡ α = | v | 2 − | v 0 | 2 . {\displaystyle 2\left|\mathbf {r} -\mathbf {r} _{0}\right|\left|\mathbf {a} \right|\cos \alpha =|\mathbf {v} |^{2}-|\mathbf {v} _{0}|^{2}.} In

9875-391: The definitions of velocity and acceleration, subject to the initial conditions r ( t 0 ) = r 0 and v ( t 0 ) = v 0 ; v = ∫ a d t = a t + v 0 , [ 1 ] r = ∫ ( a t + v 0 ) d t =

10000-1020: The direction of motion, in other words for a curved path it is the tangent vector . Loosely speaking, first order derivatives are related to tangents of curves. Still for curved paths, the acceleration is directed towards the center of curvature of the path. Again, loosely speaking, second order derivatives are related to curvature. The rotational analogues are the "angular vector" (angle the particle rotates about some axis) θ = θ ( t ) , angular velocity ω = ω ( t ) , and angular acceleration α = α ( t ) : θ = θ n ^ , ω = d θ d t , α = d ω d t , {\displaystyle {\boldsymbol {\theta }}=\theta {\hat {\mathbf {n} }}\,,\quad {\boldsymbol {\omega }}={\frac {d{\boldsymbol {\theta }}}{dt}}\,,\quad {\boldsymbol {\alpha }}={\frac {d{\boldsymbol {\omega }}}{dt}}\,,} where n̂

10125-449: The equation Δ r = v 0 t + a t 2 2 {\textstyle \Delta r=v_{0}t+{\frac {at^{2}}{2}}} . This equation is applicable when the final velocity v is unknown. It is often convenient to formulate the trajectory of a particle r ( t ) = ( x ( t ), y ( t ), z ( t )) using polar coordinates in the X – Y plane. In this case, its velocity and acceleration take

10250-460: The equation s = ⁠ 1 / 2 ⁠ gt in his work geometrically, using the Merton rule , now known as a special case of one of the equations of kinematics. Galileo was the first to show that the path of a projectile is a parabola . Galileo had an understanding of centrifugal force and gave a correct definition of momentum . This emphasis of momentum as a fundamental quantity in dynamics

10375-914: The equations expressing conservation of momentum and kinetic energy are: m A v A 1 + m B v B 1 = m A v A 2 + m B v B 2 1 2 m A v A 1 2 + 1 2 m B v B 1 2 = 1 2 m A v A 2 2 + 1 2 m B v B 2 2 . {\displaystyle {\begin{aligned}m_{A}v_{A1}+m_{B}v_{B1}&=m_{A}v_{A2}+m_{B}v_{B2}\\{\tfrac {1}{2}}m_{A}v_{A1}^{2}+{\tfrac {1}{2}}m_{B}v_{B1}^{2}&={\tfrac {1}{2}}m_{A}v_{A2}^{2}+{\tfrac {1}{2}}m_{B}v_{B2}^{2}\,.\end{aligned}}} A change of reference frame can simplify analysis of

10500-472: The equations of motion that begin to be recognized as the modern ones. Later the equations of motion also appeared in electrodynamics , when describing the motion of charged particles in electric and magnetic fields, the Lorentz force is the general equation which serves as the definition of what is meant by an electric field and magnetic field . With the advent of special relativity and general relativity ,

10625-536: The first law and the second law of motion. He did not generalize and make them applicable to bodies not subject to the earth's gravitation. That step was Newton's contribution. The term "inertia" was used by Kepler who applied it to bodies at rest. (The first law of motion is now often called the law of inertia.) Galileo did not fully grasp the third law of motion, the law of the equality of action and reaction, though he corrected some errors of Aristotle. With Stevin and others Galileo also wrote on statics. He formulated

10750-765: The following: p = ∑ i m i v i . {\displaystyle p=\sum _{i}m_{i}v_{i}.} A system of particles has a center of mass , a point determined by the weighted sum of their positions: r cm = m 1 r 1 + m 2 r 2 + ⋯ m 1 + m 2 + ⋯ = ∑ i m i r i ∑ i m i . {\displaystyle r_{\text{cm}}={\frac {m_{1}r_{1}+m_{2}r_{2}+\cdots }{m_{1}+m_{2}+\cdots }}={\frac {\sum _{i}m_{i}r_{i}}{\sum _{i}m_{i}}}.} If one or more of

10875-456: The force is between particles. Similarly, if there are several particles, the momentum exchanged between each pair of particles adds to zero, so the total change in momentum is zero. The conservation of the total momentum of a number of interacting particles can be expressed as m A v A + m B v B + m C v C + . . . = c o n s t

11000-780: The form, r ( t ) = r r ^ + z ( t ) z ^ . {\displaystyle \mathbf {r} (t)=r{\hat {\mathbf {r} }}+z(t){\hat {\mathbf {z} }}.} In general, the trajectory r ( t ) is not constrained to lie on a circular cylinder, so the radius R varies with time and the trajectory of the particle in cylindrical-polar coordinates becomes: r ( t ) = r ( t ) r ^ + z ( t ) z ^ . {\displaystyle \mathbf {r} (t)=r(t){\hat {\mathbf {r} }}+z(t){\hat {\mathbf {z} }}.} Where r , θ , and z might be continuously differentiable functions of time and

11125-972: The function notation is dropped for simplicity. The velocity vector v P is the time derivative of the trajectory r ( t ), which yields: v P = d d t ( r r ^ + z z ^ ) = v r ^ + r ω θ ^ + v z z ^ = v ( r ^ + θ ^ ) + v z z ^ . {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=v{\hat {\mathbf {r} }}+r\mathbf {\omega } {\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}=v({\hat {\mathbf {r} }}+{\hat {\mathbf {\theta } }})+v_{z}{\hat {\mathbf {z} }}.} Similarly,

11250-1047: The initial velocities are known, the final velocities are given by v A 2 = ( m A − m B m A + m B ) v A 1 + ( 2 m B m A + m B ) v B 1 v B 2 = ( m B − m A m A + m B ) v B 1 + ( 2 m A m A + m B ) v A 1 . {\displaystyle {\begin{aligned}v_{A2}&=\left({\frac {m_{A}-m_{B}}{m_{A}+m_{B}}}\right)v_{A1}+\left({\frac {2m_{B}}{m_{A}+m_{B}}}\right)v_{B1}\\v_{B2}&=\left({\frac {m_{B}-m_{A}}{m_{A}+m_{B}}}\right)v_{B1}+\left({\frac {2m_{A}}{m_{A}+m_{B}}}\right)v_{A1}\,.\end{aligned}}} If one body has much greater mass than

11375-521: The mass of a body or the forces acting upon it. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics , not kinematics. For further details, see analytical dynamics . Kinematics

11500-411: The momentum of one particle after the collision is known, the law can be used to determine the momentum of the other particle. Alternatively if the combined kinetic energy after the collision is known, the law can be used to determine the momentum of each particle after the collision. Kinetic energy is usually not conserved. If it is conserved, the collision is called an elastic collision ; if not, it

11625-433: The movement of components in a mechanical system , simplifying the derivation of the equations of motion. They are also central to dynamic analysis . Kinematic analysis is the process of measuring the kinematic quantities used to describe motion. In engineering, for instance, kinematic analysis may be used to find the range of movement for a given mechanism and, working in reverse, using kinematic synthesis to design

11750-742: The negative sign indicating that the forces oppose. Equivalently, d d t ( p 1 + p 2 ) = 0. {\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left(p_{1}+p_{2}\right)=0.} If the velocities of the particles are v A1 and v B1 before the interaction, and afterwards they are v A2 and v B2 , then m A v A 1 + m B v B 1 = m A v A 2 + m B v B 2 . {\displaystyle m_{A}v_{A1}+m_{B}v_{B1}=m_{A}v_{A2}+m_{B}v_{B2}.} This law holds no matter how complicated

11875-400: The net force is equal to the mass of the particle times its acceleration . Example : A model airplane of mass 1 kg accelerates from rest to a velocity of 6 m/s due north in 2 s. The net force required to produce this acceleration is 3  newtons due north. The change in momentum is 6 kg⋅m/s due north. The rate of change of momentum is 3 (kg⋅m/s)/s due north which

12000-438: The notation for the magnitudes of the vectors | a | = a , | v | = v , | r − r 0 | = Δ r {\displaystyle |\mathbf {a} |=a,|\mathbf {v} |=v,|\mathbf {r} -\mathbf {r} _{0}|=\Delta r} where Δ r {\displaystyle \Delta r} can be any curvaceous path taken as

12125-493: The object is constant. The results of this case are summarized below. These equations apply to a particle moving linearly, in three dimensions in a straight line with constant acceleration . Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one. v =

12250-726: The object, its velocity (the first time derivative of r , v = ⁠ d r / dt ⁠ ), and its acceleration (the second derivative of r , a = ⁠ d r / dt ⁠ ), and time t . Euclidean vectors in 3D are denoted throughout in bold. This is equivalent to saying an equation of motion in r is a second-order ordinary differential equation (ODE) in r , M [ r ( t ) , r ˙ ( t ) , r ¨ ( t ) , t ] = 0 , {\displaystyle M\left[\mathbf {r} (t),\mathbf {\dot {r}} (t),\mathbf {\ddot {r}} (t),t\right]=0\,,} where t

12375-447: The other, its velocity will be little affected by a collision while the other body will experience a large change. In an inelastic collision, some of the kinetic energy of the colliding bodies is converted into other forms of energy (such as heat or sound ). Examples include traffic collisions , in which the effect of loss of kinetic energy can be seen in the damage to the vehicles; electrons losing some of their energy to atoms (as in

12500-410: The particle's position as a function of time. The velocity of a particle is a vector quantity that describes the direction as well as the magnitude of motion of the particle. More mathematically, the rate of change of the position vector of a point with respect to time is the velocity of the point. Consider the ratio formed by dividing the difference of two positions of a particle ( displacement ) by

12625-456: The particles are numbered 1 and 2, the second law states that F 1 = ⁠ d p 1 / d t ⁠ and F 2 = ⁠ d p 2 / d t ⁠ . Therefore, d p 1 d t = − d p 2 d t , {\displaystyle {\frac {{\text{d}}p_{1}}{{\text{d}}t}}=-{\frac {{\text{d}}p_{2}}{{\text{d}}t}},} with

12750-421: The particles is moving, the center of mass of the system will generally be moving as well (unless the system is in pure rotation around it). If the total mass of the particles is m {\displaystyle m} , and the center of mass is moving at velocity v cm , the momentum of the system is: p = m v cm . {\displaystyle p=mv_{\text{cm}}.} This

12875-558: The position of point A relative to point B is the difference between their components: r A / B = r A − r B = ( x A − x B , y A − y B , z A − z B ) {\displaystyle \mathbf {r} _{A/B}=\mathbf {r} _{A}-\mathbf {r} _{B}=\left(x_{A}-x_{B},y_{A}-y_{B},z_{A}-z_{B}\right)} The velocity of one point relative to another

13000-439: The position vector | r | {\displaystyle \left|\mathbf {r} \right|} gives the distance between the point r {\displaystyle \mathbf {r} } and the origin. | r | = x 2 + y 2 + z 2 . {\displaystyle |\mathbf {r} |={\sqrt {x^{2}+y^{2}+z^{2}}}.} The direction cosines of

13125-993: The position vector provide a quantitative measure of direction. In general, an object's position vector will depend on the frame of reference; different frames will lead to different values for the position vector. The trajectory of a particle is a vector function of time, r ( t ) {\displaystyle \mathbf {r} (t)} , which defines the curve traced by the moving particle, given by r ( t ) = x ( t ) x ^ + y ( t ) y ^ + z ( t ) z ^ , {\displaystyle \mathbf {r} (t)=x(t){\hat {\mathbf {x} }}+y(t){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where x ( t ) {\displaystyle x(t)} , y ( t ) {\displaystyle y(t)} , and z ( t ) {\displaystyle z(t)} describe each coordinate of

13250-412: The principle of the parallelogram of forces, but he did not fully recognize its scope. Galileo also was interested by the laws of the pendulum, his first observations of which were as a young man. In 1583, while he was praying in the cathedral at Pisa, his attention was arrested by the motion of the great lamp lighted and left swinging, referencing his own pulse for time keeping. To him the period appeared

13375-407: The quantity of motion of a body undergoing a uniformly accelerated motion is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion. For writers on kinematics before Galileo , since small time intervals could not be measured, the affinity between time and motion was obscure. They used time as a function of distance, and in free fall, greater velocity as

13500-399: The rate of change of the magnitude of the velocity vector and the rate of change of direction of that vector. The same reasoning used with respect to the position of a particle to define velocity, can be applied to the velocity to define acceleration. The acceleration of a particle is the vector defined by the rate of change of the velocity vector. The average acceleration of a particle over

13625-402: The relevant laws of physics. Suppose x is a position in an inertial frame of reference. From the point of view of another frame of reference, moving at a constant speed u relative to the other, the position (represented by a primed coordinate) changes with time as x ′ = x − u t . {\displaystyle x'=x-ut\,.} This is called

13750-538: The resulting direction and speed of motion of objects after they collide. Below, the basic properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations (see multiple dimensions ). The momentum of a particle is conventionally represented by the letter p . It is the product of two quantities, the particle's mass (represented by the letter m ) and its velocity ( v ): p = m v . {\displaystyle p=mv.} The unit of momentum

13875-644: The same speed. Adding the speed of the center of mass to both, we find that the body that was moving is now stopped and the other is moving away at speed v . The bodies have exchanged their velocities. Regardless of the velocities of the bodies, a switch to the center of mass frame leads us to the same conclusion. Therefore, the final velocities are given by v A 2 = v B 1 v B 2 = v A 1 . {\displaystyle {\begin{aligned}v_{A2}&=v_{B1}\\v_{B2}&=v_{A1}\,.\end{aligned}}} In general, when

14000-412: The same, even after the motion had greatly diminished, discovering the isochronism of the pendulum. More careful experiments carried out by him later, and described in his Discourses, revealed the period of oscillation varies with the square root of length but is independent of the mass the pendulum. Thus we arrive at René Descartes , Isaac Newton , Gottfried Leibniz , et al.; and the evolved forms of

14125-418: The second reference frame is also an inertial frame and the accelerations are the same: a ′ = d v ′ d t = a . {\displaystyle a'={\frac {{\text{d}}v'}{{\text{d}}t}}=a\,.} Thus, momentum is conserved in both reference frames. Moreover, as long as the force has the same form, in both frames, Newton's second law

14250-403: The second time derivative of the relative position vector r B/A . Assuming that the initial conditions of the position, r 0 {\displaystyle \mathbf {r} _{0}} , and velocity v 0 {\displaystyle \mathbf {v} _{0}} at time t = 0 {\displaystyle t=0} are known, the first integration yields

14375-563: The solstices and the equinoxes of the Sun and the period of the Moon . But they had nothing other than a set of algorithms to guide them. Equations of motion were not written down for another thousand years. Medieval scholars in the thirteenth century — for example at the relatively new universities in Oxford and Paris — drew on ancient mathematicians (Euclid and Archimedes) and philosophers (Aristotle) to develop

14500-410: The system for all times t after t = 0 . Other dynamical variables like the momentum p of the object, or quantities derived from r and p like angular momentum , can be used in place of r as the quantity to solve for from some equation of motion, although the position of the object at time t is by far the most sought-after quantity. Sometimes, the equation will be linear and

14625-514: The theoretical modifications to spacetime meant the classical equations of motion were also modified to account for the finite speed of light , and curvature of spacetime . In all these cases the differential equations were in terms of a function describing the particle's trajectory in terms of space and time coordinates, as influenced by forces or energy transformations. However, the equations of quantum mechanics can also be considered "equations of motion", since they are differential equations of

14750-404: The time derivative of the relative position vector r B/A . The acceleration of one point C relative to another point B is simply the difference between their accelerations. a C / B = a C − a B {\displaystyle \mathbf {a} _{C/B}=\mathbf {a} _{C}-\mathbf {a} _{B}} which is the difference between

14875-1152: The time interval. This ratio is called the average velocity over that time interval and is defined as v ¯ = Δ r Δ t = Δ x Δ t x ^ + Δ y Δ t y ^ + Δ z Δ t z ^ = v ¯ x x ^ + v ¯ y y ^ + v ¯ z z ^ {\displaystyle \mathbf {\bar {v}} ={\frac {\Delta \mathbf {r} }{\Delta t}}={\frac {\Delta x}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta y}{\Delta t}}{\hat {\mathbf {y} }}+{\frac {\Delta z}{\Delta t}}{\hat {\mathbf {z} }}={\bar {v}}_{x}{\hat {\mathbf {x} }}+{\bar {v}}_{y}{\hat {\mathbf {y} }}+{\bar {v}}_{z}{\hat {\mathbf {z} }}\,} where Δ r {\displaystyle \Delta \mathbf {r} }

15000-471: The top area (a triangle). The area of a triangle is 1 2 B H {\textstyle {\frac {1}{2}}BH} where B {\displaystyle B} is the base and H {\displaystyle H} is the height. In this case, B = t {\displaystyle B=t} and H = a t {\displaystyle H=at} or A = 1 2 B H = 1 2

15125-481: The tower is 50 m high, and this height is measured along the z -axis, then the coordinate vector to the top of the tower is r = (0 m, −50 m, 50 m). In the most general case, a three-dimensional coordinate system is used to define the position of a particle. However, if the particle is constrained to move within a plane, a two-dimensional coordinate system is sufficient. All observations in physics are incomplete without being described with respect to

15250-803: The velocity and acceleration vectors simplify. The velocity of v P is the time derivative of the trajectory r ( t ), v P = d d t ( r r ^ + z z ^ ) = r ω θ ^ + v z z ^ = v θ ^ + v z z ^ . {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=r\omega {\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}=v{\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}.} A special case of

15375-1543: The velocity increases linearly, so the average velocity multiplied by time is the distance traveled while increasing the velocity from v 0 to v , as can be illustrated graphically by plotting velocity against time as a straight line graph. Algebraically, it follows from solving [1] for a = ( v − v 0 ) t {\displaystyle \mathbf {a} ={\frac {(\mathbf {v} -\mathbf {v} _{0})}{t}}} and substituting into [2] r = r 0 + v 0 t + t 2 ( v − v 0 ) , {\displaystyle \mathbf {r} =\mathbf {r} _{0}+\mathbf {v} _{0}t+{\frac {t}{2}}(\mathbf {v} -\mathbf {v} _{0})\,,} then simplifying to get r = r 0 + t 2 ( v + v 0 ) {\displaystyle \mathbf {r} =\mathbf {r} _{0}+{\frac {t}{2}}(\mathbf {v} +\mathbf {v} _{0})} or in magnitudes r = r 0 + ( v + v 0 2 ) t [ 3 ] {\displaystyle r=r_{0}+\left({\frac {v+v_{0}}{2}}\right)t\quad [3]} From [3], t = ( r − r 0 ) ( 2 v + v 0 ) {\displaystyle t=\left(r-r_{0}\right)\left({\frac {2}{v+v_{0}}}\right)} Kinematics Kinematics

15500-676: The velocity of point A relative to point B is the difference between their components: v A / B = v A − v B = ( v A x − v B x , v A y − v B y , v A z − v B z ) {\displaystyle \mathbf {v} _{A/B}=\mathbf {v} _{A}-\mathbf {v} _{B}=\left(v_{A_{x}}-v_{B_{x}},v_{A_{y}}-v_{B_{y}},v_{A_{z}}-v_{B_{z}}\right)} Alternatively, this same result could be obtained by computing

15625-683: The velocity of the particle as a function of time. v ( t ) = v 0 + ∫ 0 t a d τ = v 0 + a t . {\displaystyle \mathbf {v} (t)=\mathbf {v} _{0}+\int _{0}^{t}\mathbf {a} \,{\text{d}}\tau =\mathbf {v} _{0}+\mathbf {a} t.} A second integration yields its path (trajectory), r ( t ) = r 0 + ∫ 0 t v ( τ ) d τ = r 0 + ∫ 0 t ( v 0 +

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