Misplaced Pages

#448551

61-434: Collision is short-duration interaction between two bodies or more than two bodies simultaneously causing change in motion of bodies involved due to internal forces acted between them during this. Collisions involve forces (there is a change in velocity ). The magnitude of the velocity difference just before impact is called the closing speed . All collisions conserve momentum . What distinguishes different types of collisions

122-491: A ⋅ x ) {\displaystyle \therefore v^{2}=u^{2}+2({\boldsymbol {a}}\cdot {\boldsymbol {x}})} where v = | v | etc. The above equations are valid for both Newtonian mechanics and special relativity . Where Newtonian mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in Newtonian mechanics, all observers agree on

183-605: A ) ⋅ x = ( 2 a ) ⋅ ( u t + 1 2 a t 2 ) = 2 t ( a ⋅ u ) + a 2 t 2 = v 2 − u 2 {\displaystyle (2{\boldsymbol {a}})\cdot {\boldsymbol {x}}=(2{\boldsymbol {a}})\cdot ({\boldsymbol {u}}t+{\tfrac {1}{2}}{\boldsymbol {a}}t^{2})=2t({\boldsymbol {a}}\cdot {\boldsymbol {u}})+a^{2}t^{2}=v^{2}-u^{2}} ∴ v 2 = u 2 + 2 (

244-447: A = d v d t . {\displaystyle {\boldsymbol {a}}={\frac {d{\boldsymbol {v}}}{dt}}.} From there, velocity is expressed as the area under an a ( t ) acceleration vs. time graph. As above, this is done using the concept of the integral: v = ∫ a   d t . {\displaystyle {\boldsymbol {v}}=\int {\boldsymbol {a}}\ dt.} In

305-439: A combination of these mechanisms. Railguns utilize electromagnetic fields to provide a constant acceleration along the entire length of the device, greatly increasing the muzzle velocity . Some projectiles provide propulsion during flight by means of a rocket engine or jet engine . In military terminology, a rocket is unguided, while a missile is guided . Note the two meanings of "rocket" (weapon and engine): an ICBM

366-417: A constant 20 kilometres per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes. Hence, the car is considered to be undergoing an acceleration. Since the derivative of the position with respect to time gives the change in position (in metres ) divided by the change in time (in seconds ), velocity is measured in metres per second (m/s). Velocity

427-687: A specific angle θ {\displaystyle \theta } : 1. Time to reach maximum height. It is symbolized as ( t {\displaystyle t} ), which is the time taken for the projectile to reach the maximum height from the plane of projection. Mathematically, it is given as t = U sin ⁡ θ / g {\displaystyle t=U\sin \theta /g} where g {\displaystyle g} = acceleration due to gravity (app 9.81 m/s²), U {\displaystyle U} = initial velocity (m/s) and θ {\displaystyle \theta } = angle made by

488-588: A two-dimensional system, where there is an x-axis and a y-axis, corresponding velocity components are defined as v x = d x / d t , {\displaystyle v_{x}=dx/dt,} v y = d y / d t . {\displaystyle v_{y}=dy/dt.} The two-dimensional velocity vector is then defined as v =< v x , v y > {\displaystyle {\textbf {v}}=<v_{x},v_{y}>} . The magnitude of this vector represents speed and

549-402: A velocity vector, denotes only how fast an object is moving, while velocity indicates both an object's speed and direction. To have a constant velocity , an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed. For example, a car moving at

610-791: A zero-friction collision of a moving ball with a stationary one of equal mass, the angle between the directions of the two balls is 90 degrees. This is an important fact that professional billiards players take into account, although it assumes the ball is moving without any impact of friction across the table rather than rolling with friction. Consider an elastic collision in two dimensions of any two masses m 1 and m 2 , with respective initial velocities u 1 and u 2 where u 2 = 0 , and final velocities V 1 and V 2 . Conservation of momentum gives m 1 u 1 = m 1 V 1 + m 2 V 2 . Conservation of energy for an elastic collision gives (1/2) m 1 | u 1 | = (1/2) m 1 | V 1 | + (1/2) m 2 | V 2 |. Now consider

671-508: Is a projectile weapon based solely on a projectile's kinetic energy to inflict damage to a target, instead of using any explosive , incendiary / thermal , chemical or radiological payload . All kinetic weapons work by attaining a high flight speed — generally supersonic or even up to hypervelocity — and collide with their targets, converting their kinetic energy and relative impulse into destructive shock waves , heat and cavitation . In kinetic weapons with unpowered flight ,

SECTION 10

#1732851174449

732-468: Is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. The drag force, F D {\displaystyle F_{D}} , is dependent on the square of velocity and is given as F D = 1 2 ρ v 2 C D A {\displaystyle F_{D}\,=\,{\tfrac {1}{2}}\,\rho \,v^{2}\,C_{D}\,A} where Escape velocity

793-429: Is a guided missile with a rocket engine. An explosion, whether or not by a weapon, causes the debris to act as multiple high velocity projectiles. An explosive weapon or device may also be designed to produce many high velocity projectiles by the break-up of its casing; these are correctly termed fragments . In projectile motion the most important force applied to the ‘projectile’ is the propelling force, in this case

854-417: Is always less than or equal to the average speed of an object. This can be seen by realizing that while distance is always strictly increasing, displacement can increase or decrease in magnitude as well as change direction. In terms of a displacement-time ( x vs. t ) graph, the instantaneous velocity (or, simply, velocity) can be thought of as the slope of the tangent line to the curve at any point , and

915-745: Is an object that is propelled by the application of an external force and then moves freely under the influence of gravity and air resistance . Although any objects in motion through space are projectiles, they are commonly found in warfare and sports (for example, a thrown baseball , kicked football , fired bullet , shot arrow , stone released from catapult ). In ballistics mathematical equations of motion are used to analyze projectile trajectories through launch, flight , and impact . Blowguns and pneumatic rifles use compressed gases, while most other guns and cannons utilize expanding gases liberated by sudden chemical reactions by propellants like smokeless powder . Light-gas guns use

976-720: Is defined as v =< v x , v y , v z > {\displaystyle {\textbf {v}}=<v_{x},v_{y},v_{z}>} with its magnitude also representing speed and being determined by | v | = v x 2 + v y 2 + v z 2 . {\displaystyle |v|={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}.} While some textbooks use subscript notation to define Cartesian components of velocity, others use u {\displaystyle u} , v {\displaystyle v} , and w {\displaystyle w} for

1037-501: Is defined as the rate of change of position with respect to time, which may also be referred to as the instantaneous velocity to emphasize the distinction from the average velocity. In some applications the average velocity of an object might be needed, that is to say, the constant velocity that would provide the same resultant displacement as a variable velocity in the same time interval, v ( t ) , over some time period Δ t . Average velocity can be calculated as: The average velocity

1098-435: Is defined as the rate of change of position, it is often common to start with an expression for an object's acceleration . As seen by the three green tangent lines in the figure, an object's instantaneous acceleration at a point in time is the slope of the line tangent to the curve of a v ( t ) graph at that point. In other words, instantaneous acceleration is defined as the derivative of velocity with respect to time:

1159-483: Is found by the distance formula as | v | = v x 2 + v y 2 . {\displaystyle |v|={\sqrt {v_{x}^{2}+v_{y}^{2}}}.} In three-dimensional systems where there is an additional z-axis, the corresponding velocity component is defined as v z = d z / d t . {\displaystyle v_{z}=dz/dt.} The three-dimensional velocity vector

1220-635: Is given by the harmonic mean of the speeds v ¯ = n ( 1 v 1 + 1 v 2 + 1 v 3 + ⋯ + 1 v n ) − 1 = n ( ∑ i = 1 n 1 v i ) − 1 . {\displaystyle {\bar {v}}=n\left({1 \over v_{1}}+{1 \over v_{2}}+{1 \over v_{3}}+\dots +{1 \over v_{n}}\right)^{-1}=n\left(\sum _{i=1}^{n}{\frac {1}{v_{i}}}\right)^{-1}.} Although velocity

1281-467: Is known as moment of inertia . If forces are in the radial direction only with an inverse square dependence, as in the case of a gravitational orbit , angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion . Projectile A projectile

SECTION 20

#1732851174449

1342-498: Is measured in the SI ( metric system ) as metres per second (m/s or m⋅s ). For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a vector. If there is a change in speed, direction or both, then the object is said to be undergoing an acceleration . The average velocity of an object over a period of time is its change in position , Δ s {\displaystyle \Delta s} , divided by

1403-474: Is position and r ^ {\displaystyle {\hat {\boldsymbol {r}}}} is the radial direction. The transverse speed (or magnitude of the transverse velocity) is the magnitude of the cross product of the unit vector in the radial direction and the velocity vector. It is also the dot product of velocity and transverse direction, or the product of the angular speed ω {\displaystyle \omega } and

1464-506: Is quantifying the forces generated during the foot-ground collisions associated with both disabled and non-disabled gait. This quantification typically requires subjects to walk across a force platform (sometimes called a "force plate") as well as detailed kinematic and dynamic (sometimes termed kinetic) analysis. Hypervelocity is very high velocity , approximately over 3,000 meters per second (11,000 km/h, 6,700 mph, 10,000 ft/s, or Mach 8.8). In particular, hypervelocity

1525-409: Is the gravitational constant and g is the gravitational acceleration . The escape velocity from Earth's surface is about 11 200 m/s, and is irrespective of the direction of the object. This makes "escape velocity" somewhat of a misnomer, as the more correct term would be "escape speed": any object attaining a velocity of that magnitude, irrespective of atmosphere, will leave the vicinity of

1586-422: Is the speed in combination with the direction of motion of an object . Velocity is a fundamental concept in kinematics , the branch of classical mechanics that describes the motion of bodies. Velocity is a physical vector quantity : both magnitude and direction are needed to define it. The scalar absolute value ( magnitude ) of velocity is called speed , being a coherent derived unit whose quantity

1647-769: Is the component of velocity along a circle centered at the origin. v = v T + v R {\displaystyle {\boldsymbol {v}}={\boldsymbol {v}}_{T}+{\boldsymbol {v}}_{R}} where The radial speed (or magnitude of the radial velocity) is the dot product of the velocity vector and the unit vector in the radial direction. v R = v ⋅ r | r | = v ⋅ r ^ {\displaystyle v_{R}={\frac {{\boldsymbol {v}}\cdot {\boldsymbol {r}}}{\left|{\boldsymbol {r}}\right|}}={\boldsymbol {v}}\cdot {\hat {\boldsymbol {r}}}} where r {\displaystyle {\boldsymbol {r}}}

1708-415: Is the line that is collinear to the common normal of the surfaces that are closest or in contact during impact. This is the line along which internal force of collision acts during impact, and Newton's coefficient of restitution is defined only along this line. Collisions in ideal gases approach perfectly elastic collisions, as do scattering interactions of sub-atomic particles which are deflected by

1769-418: Is the mass of the object. The kinetic energy of a moving object is dependent on its velocity and is given by the equation E k = 1 2 m v 2 {\displaystyle E_{\text{k}}={\tfrac {1}{2}}mv^{2}} where E k is the kinetic energy. Kinetic energy is a scalar quantity as it depends on the square of the velocity. In fluid dynamics , drag

1830-461: Is the mass times the distance to the origin times the transverse velocity, or equivalently, the mass times the distance squared times the angular speed. The sign convention for angular momentum is the same as that for angular velocity. L = m r v T = m r 2 ω {\displaystyle L=mrv_{T}=mr^{2}\omega } where The expression m r 2 {\displaystyle mr^{2}}

1891-433: Is the maximum height attained by the projectile OR the maximum displacement on the vertical axis (y-axis) covered by the projectile. It is given as H = U 2 sin 2 ⁡ θ / 2 g {\displaystyle H=U^{2}\sin ^{2}\theta /2g} . 4. Range ( R {\displaystyle R} ): The Range of a projectile is the horizontal distance covered (on

Collision - Misplaced Pages Continue

1952-547: Is the minimum speed a ballistic object needs to escape from a massive body such as Earth. It represents the kinetic energy that, when added to the object's gravitational potential energy (which is always negative), is equal to zero. The general formula for the escape velocity of an object at a distance r from the center of a planet with mass M is v e = 2 G M r = 2 g r , {\displaystyle v_{\text{e}}={\sqrt {\frac {2GM}{r}}}={\sqrt {2gr}},} where G

2013-458: Is the rate of rotation about the origin (with positive quantities representing counter-clockwise rotation and negative quantities representing clockwise rotation, in a right-handed coordinate system). The radial and traverse velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. The transverse velocity

2074-453: Is the speed of light. Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. Relative velocity is fundamental in both classical and modern physics, since many systems in physics deal with the relative motion of two or more particles. Consider an object A moving with velocity vector v and an object B with velocity vector w ; these absolute velocities are typically expressed in

2135-525: Is velocity so high that the strength of materials upon impact is very small compared to inertial stresses. Thus, metals and fluids behave alike under hypervelocity impact. An impact under extreme hypervelocity results in vaporization of the impactor and target. For structural metals, hypervelocity is generally considered to be over 2,500 m/s (5,600 mph, 9,000 km/h, 8,200 ft/s, or Mach 7.3). Meteorite craters are also examples of hypervelocity impacts. Velocity Velocity

2196-448: Is whether they also conserve kinetic energy of the system before and after the collision. Collisions are of three types: The degree to which a collision is elastic or inelastic is quantified by the coefficient of restitution , a value that generally ranges between zero and one. A perfectly elastic collision has a coefficient of restitution of one; a perfectly inelastic collision has a coefficient of restitution of zero. The line of impact

2257-436: The x {\displaystyle x} -, y {\displaystyle y} -, and z {\displaystyle z} -axes respectively. In polar coordinates , a two-dimensional velocity is described by a radial velocity , defined as the component of velocity away from or toward the origin, and a transverse velocity , perpendicular to the radial one. Both arise from angular velocity , which

2318-424: The derivative of the position with respect to time: v = lim Δ t → 0 Δ s Δ t = d s d t . {\displaystyle {\boldsymbol {v}}=\lim _{{\Delta t}\to 0}{\frac {\Delta {\boldsymbol {s}}}{\Delta t}}={\frac {d{\boldsymbol {s}}}{dt}}.} From this derivative equation, in

2379-440: The electromagnetic force . Some large-scale interactions like the slingshot type gravitational interactions between satellites and planets are almost perfectly elastic. Collisions play an important role in cue sports . Because the collisions between billiard balls are nearly elastic , and the balls roll on a surface that produces low rolling friction , their behavior is often used to illustrate Newton's laws of motion . After

2440-724: The muzzle velocity or launch velocity often determines the effective range and potential damage of the kinetic projectile. Kinetic weapons are the oldest and most common ranged weapons used in human history , with the projectiles varying from blunt projectiles such as rocks and round shots , pointed missiles such as arrows , bolts , darts , and javelins , to modern tapered high-velocity impactors such as bullets , flechettes , and penetrators . Typical kinetic weapons accelerate their projectiles mechanically (by muscle power , mechanical advantage devices , elastic energy or pneumatics ) or chemically (by propellant combustion , as with firearms ), but newer technologies are enabling

2501-1574: The average velocity as the slope of the secant line between two points with t coordinates equal to the boundaries of the time period for the average velocity. If t 1 = t 2 = t 3 = ... = t , then average speed is given by the arithmetic mean of the speeds v ¯ = v 1 + v 2 + v 3 + ⋯ + v n n = 1 n ∑ i = 1 n v i {\displaystyle {\bar {v}}={v_{1}+v_{2}+v_{3}+\dots +v_{n} \over n}={\frac {1}{n}}\sum _{i=1}^{n}{v_{i}}} v ¯ = s 1 + s 2 + s 3 + ⋯ + s n t 1 + t 2 + t 3 + ⋯ + t n = s 1 + s 2 + s 3 + ⋯ + s n s 1 v 1 + s 2 v 2 + s 3 v 3 + ⋯ + s n v n {\displaystyle {\bar {v}}={s_{1}+s_{2}+s_{3}+\dots +s_{n} \over t_{1}+t_{2}+t_{3}+\dots +t_{n}}={{s_{1}+s_{2}+s_{3}+\dots +s_{n}} \over {{s_{1} \over v_{1}}+{s_{2} \over v_{2}}+{s_{3} \over v_{3}}+\dots +{s_{n} \over v_{n}}}}} If s 1 = s 2 = s 3 = ... = s , then average speed

Collision - Misplaced Pages Continue

2562-451: The base body as long as it does not intersect with something in its path. In special relativity , the dimensionless Lorentz factor appears frequently, and is given by γ = 1 1 − v 2 c 2 {\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}} where γ is the Lorentz factor and c

2623-425: The case m 1 = m 2 : we obtain u 1 = V 1 + V 2 and | u 1 | = | V 1 | + | V 2 |. Taking the dot product of each side of the former equation with itself, | u 1 | = u 1 • u 1 = | V 1 | + | V 2 | + 2 V 1 • V 2 . Comparing this with the latter equation gives V 1 • V 2 = 0, so they are perpendicular unless V 1 is the zero vector (which occurs if and only if

2684-421: The collision is head-on). [REDACTED] In a perfect inelastic collision , i.e., a zero coefficient of restitution , the colliding particles coalesce . It is necessary to consider conservation of momentum: where v is the final velocity, which is hence given by The reduction of total kinetic energy is equal to the total kinetic energy before the collision in a center of momentum frame with respect to

2745-402: The concept of an instantaneous velocity might at first seem counter-intuitive, it may be thought of as the velocity that the object would continue to travel at if it stopped accelerating at that moment. While the terms speed and velocity are often colloquially used interchangeably to connote how fast an object is moving, in scientific terms they are different. Speed, the scalar magnitude of

2806-408: The development of potential weapons using electromagnetically launched projectiles, such as railguns , coilguns and mass drivers . There are also concept weapons that are accelerated by gravity , as in the case of kinetic bombardment weapons designed for space warfare . Some projectiles stay connected by a cable to the launch equipment after launching it: An object projected at an angle to

2867-458: The duration of the period, Δ t {\displaystyle \Delta t} , given mathematically as v ¯ = Δ s Δ t . {\displaystyle {\bar {v}}={\frac {\Delta s}{\Delta t}}.} The instantaneous velocity of an object is the limit average velocity as the time interval approaches zero. At any particular time t , it can be calculated as

2928-481: The horizontal has both the vertical and horizontal components of velocity. The vertical component of the velocity on the y-axis is given as V y = U sin ⁡ θ {\displaystyle V_{y}=U\sin \theta } while the horizontal component of the velocity is V x = U cos ⁡ θ {\displaystyle V_{x}=U\cos \theta } . There are various calculations for projectiles at

2989-540: The inertial frame chosen is that in which the latter of the two mentioned objects is in rest. In Newtonian mechanics, the relative velocity is independent of the chosen inertial reference frame. This is not the case anymore with special relativity in which velocities depend on the choice of reference frame. In the one-dimensional case, the velocities are scalars and the equation is either: v rel = v − ( − w ) , {\displaystyle v_{\text{rel}}=v-(-w),} if

3050-477: The one-dimensional case it can be seen that the area under a velocity vs. time ( v vs. t graph) is the displacement, s . In calculus terms, the integral of the velocity function v ( t ) is the displacement function s ( t ) . In the figure, this corresponds to the yellow area under the curve. s = ∫ v   d t . {\displaystyle {\boldsymbol {s}}=\int {\boldsymbol {v}}\ dt.} Although

3111-447: The projectile with the horizontal axis. 2. Time of flight ( T {\displaystyle T} ): this is the total time taken for the projectile to fall back to the same plane from which it was projected. Mathematically it is given as T = 2 U sin ⁡ θ / g {\displaystyle T=2U\sin \theta /g} . 3. Maximum Height ( H {\displaystyle H} ): this

SECTION 50

#1732851174449

3172-648: The propelling forces are the muscles that act upon the ball to make it move, and the stronger the force applied, the more propelling force, which means the projectile (the ball) will travel farther. See pitching , bowling . Many projectiles, e.g. shells , may carry an explosive charge or another chemical or biological substance. Aside from explosive payload, a projectile can be designed to cause special damage, e.g. fire (see also early thermal weapons ), or poisoning (see also arrow poison ). A kinetic energy weapon (also known as kinetic weapon, kinetic energy warhead, kinetic warhead, kinetic projectile, kinetic kill vehicle)

3233-754: The radius (the magnitude of the position). v T = | r × v | | r | = v ⋅ t ^ = ω | r | {\displaystyle v_{T}={\frac {|{\boldsymbol {r}}\times {\boldsymbol {v}}|}{|{\boldsymbol {r}}|}}={\boldsymbol {v}}\cdot {\hat {\boldsymbol {t}}}=\omega |{\boldsymbol {r}}|} such that ω = | r × v | | r | 2 . {\displaystyle \omega ={\frac {|{\boldsymbol {r}}\times {\boldsymbol {v}}|}{|{\boldsymbol {r}}|^{2}}}.} Angular momentum in scalar form

3294-710: The same inertial reference frame . Then, the velocity of object A relative to object B is defined as the difference of the two velocity vectors: v A  relative to  B = v − w {\displaystyle {\boldsymbol {v}}_{A{\text{ relative to }}B}={\boldsymbol {v}}-{\boldsymbol {w}}} Similarly, the relative velocity of object B moving with velocity w , relative to object A moving with velocity v is: v B  relative to  A = w − v {\displaystyle {\boldsymbol {v}}_{B{\text{ relative to }}A}={\boldsymbol {w}}-{\boldsymbol {v}}} Usually,

3355-463: The situation of two objects pushed away from each other, e.g. shooting a projectile , or a rocket applying thrust (compare the derivation of the Tsiolkovsky rocket equation ). Collisions of an animal's foot or paw with the underlying substrate are generally termed ground reaction forces. These collisions are inelastic, as kinetic energy is not conserved. An important research topic in prosthetics

3416-432: The special case of constant acceleration, velocity can be studied using the suvat equations . By considering a as being equal to some arbitrary constant vector, this shows v = u + a t {\displaystyle {\boldsymbol {v}}={\boldsymbol {u}}+{\boldsymbol {a}}t} with v as the velocity at time t and u as the velocity at time t = 0 . By combining this equation with

3477-415: The suvat equation x = u t + a t /2 , it is possible to relate the displacement and the average velocity by x = ( u + v ) 2 t = v ¯ t . {\displaystyle {\boldsymbol {x}}={\frac {({\boldsymbol {u}}+{\boldsymbol {v}})}{2}}t={\boldsymbol {\bar {v}}}t.} It is also possible to derive an expression for

3538-443: The system of two particles, because in such a frame the kinetic energy after the collision is zero. In this frame most of the kinetic energy before the collision is that of the particle with the smaller mass. In another frame, in addition to the reduction of kinetic energy there may be a transfer of kinetic energy from one particle to the other; the fact that this depends on the frame shows how relative this is. With time reversed we have

3599-401: The two objects are moving in opposite directions, or: v rel = v − ( + w ) , {\displaystyle v_{\text{rel}}=v-(+w),} if the two objects are moving in the same direction. In multi-dimensional Cartesian coordinate systems , velocity is broken up into components that correspond with each dimensional axis of the coordinate system. In

3660-555: The value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. In other words, only relative velocity can be calculated. In classical mechanics, Newton's second law defines momentum , p, as a vector that is the product of an object's mass and velocity, given mathematically as p = m v {\displaystyle {\boldsymbol {p}}=m{\boldsymbol {v}}} where m

3721-648: The velocity independent of time, known as the Torricelli equation , as follows: v 2 = v ⋅ v = ( u + a t ) ⋅ ( u + a t ) = u 2 + 2 t ( a ⋅ u ) + a 2 t 2 {\displaystyle v^{2}={\boldsymbol {v}}\cdot {\boldsymbol {v}}=({\boldsymbol {u}}+{\boldsymbol {a}}t)\cdot ({\boldsymbol {u}}+{\boldsymbol {a}}t)=u^{2}+2t({\boldsymbol {a}}\cdot {\boldsymbol {u}})+a^{2}t^{2}} ( 2

SECTION 60

#1732851174449
#448551