The velocity of an object is the rate of change of its position with
respect to a frame of reference, and is a function of time. Velocity
is equivalent to a specification of its speed and direction of motion
(e.g. 7001600000000000000♠60 km/h to the north).
Contents 1 Constant velocity vs acceleration 2 Distinction between speed and velocity 3 Equation of motion 3.1 Average velocity 3.2 Instantaneous velocity 3.3 Relationship to acceleration 3.3.1 Constant acceleration 3.4 Quantities that are dependent on velocity 4 Relative velocity 4.1 Scalar velocities 5 Polar coordinates 6 See also 7 Notes 8 References 9 External links Constant velocity vs acceleration 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 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. Distinction between speed and velocity Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.
v ¯ = Δ x Δ t . displaystyle boldsymbol bar v = frac Delta boldsymbol x Delta mathit t . The average velocity 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 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. The average velocity is the same as the velocity averaged over time – that is to say, its time-weighted average, which may be calculated as the time integral of the velocity: v ¯ = 1 t 1 − t 0 ∫ t 0 t 1 v ( t ) d t , displaystyle boldsymbol bar v = 1 over t_ 1 -t_ 0 int _ t_ 0 ^ t_ 1 boldsymbol v (t) dt, where we may identify Δ x = ∫ t 0 t 1 v ( t ) d t displaystyle Delta boldsymbol x =int _ t_ 0 ^ t_ 1 boldsymbol v (t) dt and Δ t = t 1 − t 0 . displaystyle Delta t=t_ 1 -t_ 0 . Instantaneous velocity Example of a velocity vs. time graph, and the relationship between velocity v on the y-axis, acceleration a (the three green tangent lines represent the values for acceleration at different points along the curve) and displacement s (the yellow area under the curve.) If we consider v as velocity and x as the displacement (change in position) vector, then we can express the (instantaneous) velocity of a particle or object, at any particular time t, as the derivative of the position with respect to time: v = lim Δ t → 0 Δ x Δ t = d x d t . displaystyle boldsymbol v =lim _ Delta t to 0 frac Delta boldsymbol x Delta t = frac d boldsymbol x d mathit t . From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time (v vs. t graph) is the displacement, x. In calculus terms, the integral of the velocity function v(t) is the displacement function x(t). In the figure, this corresponds to the yellow area under the curve labeled s (s being an alternative notation for displacement). x = ∫ v d t . displaystyle boldsymbol x =int boldsymbol v d mathit t . 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). Although 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. Relationship to acceleration Although velocity 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, acceleration is defined as the derivative of velocity with respect to time: a = d v d t . displaystyle boldsymbol a = frac d boldsymbol v d mathit t . From there, we can obtain an expression for velocity 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 d mathit t . Constant acceleration In 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, it is trivial to show that 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 the suvat equation x = ut + at2/2, 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 mathit t = boldsymbol bar v mathit t . It is also possible to derive an expression for 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 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+ frac 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 ( 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
E k = 1 2 m v 2 displaystyle E_ text k = tfrac 1 2 mv^ 2 ignoring special relativity, where Ek is the kinetic energy and m is
the mass.
p = m v displaystyle boldsymbol p =m boldsymbol v In special relativity, the dimensionless
γ = 1 1 − v 2 c 2 displaystyle gamma = frac 1 sqrt 1- frac v^ 2 c^ 2 where γ is the
v e = 2 G M r = 2 g r , displaystyle v_ text e = sqrt frac 2GM r = sqrt 2gr , where G is the
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 the inertial frame is chosen in which the latter of the two mentioned objects is in rest. Scalar velocities In the one-dimensional case,[2] the velocities are scalars and the equation is either: v r e l = v − ( − w ) displaystyle ,v_ rel =v-(-w) , if the two objects are moving in opposite directions, or: v r e l = v − ( + w ) displaystyle ,v_ rel =v-(+w) , if the two objects are moving in the same direction. Polar coordinates 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 (also known as velocity made good), and an angular velocity, which 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 angular velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. The transverse velocity 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 v T displaystyle boldsymbol v _ T is the transverse velocity v R displaystyle boldsymbol v _ R is the radial velocity. The magnitude of the radial velocity is the dot product of the velocity vector and the unit vector in the direction of the displacement. v R = v ⋅ r
r
displaystyle v_ R = frac boldsymbol v cdot boldsymbol r left boldsymbol r right where r displaystyle boldsymbol r is displacement. The magnitude of the transverse velocity is that of the cross product of the unit vector in the direction of the displacement and the velocity vector. It is also the product of the angular speed ω displaystyle omega and the magnitude of the displacement. v T =
r × v
r
= ω
r
displaystyle v_ T = frac boldsymbol r times boldsymbol v boldsymbol r =omega boldsymbol r such that ω =
r × v
r
2 . displaystyle omega = frac boldsymbol r times boldsymbol v boldsymbol r ^ 2 .
L = m r v T = m r 2 ω displaystyle L=mrv_ T =mr^ 2 omega , where m displaystyle m, is mass r = ‖ r ‖ . displaystyle r= boldsymbol r . The expression m r 2 displaystyle mr^ 2 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. See also
Notes ^ Wilson, Edwin Bidwell (1901). Vector analysis: a text-book for the use of students of mathematics and physics, founded upon the lectures of J. Willard Gibbs. p. 125. This is the likely origin of the speed/velocity terminology in vector physics. ^ Basic principle References Robert Resnick and Jearl Walker, Fundamentals of Physics, Wiley; 7 Sub edition (June 16, 2004). ISBN 0-471-23231-9. External links Wikimedia Commons has media related to Velocity. physicsabout.com,
v t e Kinematics ← Integrate … Differentiate → Absement
Displacement (Distance)
v t e
Linear/translational quantities Angular/rotational quantities Dimensions 1 L L2 Dimensions 1 1 1 T time: t s absement: A m s T time: t s 1 distance: d, position: r, s, x, displacement m area: A m2 1 angle: θ, angular displacement: θ rad solid angle: Ω rad2, sr T−1 frequency: f s−1, Hz speed: v, velocity: v m s−1 kinematic viscosity: ν, specific angular momentum: h m2 s−1 T−1 frequency: f s−1, Hz angular speed: ω, angular velocity: ω rad s−1 T−2 acceleration: a m s−2 T−2 angular acceleration: α rad s−2 T−3 jerk: j m s−3 T−3 angular jerk: ζ rad s−3 M mass: m kg ML2 moment of inertia: I kg m2 MT−1 momentum: p, impulse: J kg m s−1, N s action: 𝒮, actergy: ℵ kg m2 s−1, J s ML2T−1 angular momentum: L, angular impulse: ΔL kg m2 s−1 action: 𝒮, actergy: ℵ kg m2 s−1, J s MT−2 force: F, weight: Fg kg m s−2, N energy: E, work: W kg m2 s−2, J ML2T−2 torque: τ, moment: M kg m2 s−2, N m energy: E, work: W kg m2 s−2, J MT−3 yank: Y kg m s−3, N s−1 power: P kg m2 s−3, W ML2T−3 rotatum: P kg m2 s−3, N m s−1 power: P kg m2 s |