Velocity

The velocity of an object is simply its speed in a particular direction. Note that both speed and direction are required to define a velocity.

Explanation

The velocity (v) is an physical quantity of the motion.

A change in an object's velocity can therefore arise from either a change in its speed or in its direction. For example an aeroplane that is circling at a constant speed of 200 km/h is changing its velocity because it is continously changing its direction. A aeroplane that is taking-off may go from zero to 200 km/h in a straight line and so would also be changing its velocity.

A change in velocity is called an acceleration. Objects are only accelerated if a force is applied to them. (The amount of acceleration depends the size of the force and the mass of the object being shifted, see Newton's Second Law of Motion.) In the case of the circling aeroplane, the pilot banks to use the force of lift from the wings to change direction. In another example the Space Shuttle orbits the earth at a constant speed but is constantly changing its velocity because of the circular orbit. In this case the force causing the acceleration is provided by the earth's gravity acting on the shuttle.

The average speed v of an object moving a distance d during a time interval t is described by the formula:

$v={\frac {d}{t}}$

Acceleration is the rate of change of an object's velocity over time. The average acceleration of a of an object whose speed changes from v_i to v_f during a time interval t is given by:

$a={\frac {(v_{f}-v_{i})}{t}}$

 Where $v_i$
= an object's initial velocity and $v_f$
= the object's final velocity over a period of time t

Formal description

Velocity (symbol: v) is a vector measurement of the rate and direction of motion. The scalar absolute value (magnitude) of velocity is speed. Velocity can also be defined as rate of change of displacement or just as the rate of displacement, depending on how the term displacement is used. It is thus a vector quantity with dimension length/time. In the SI (metric) system it is measured in metre per second

The instantaneous velocity vector v of an object that has position at time t is given by x(t) can be computed as the derivative

$v={\mathrm {d} x \over \mathrm {d} t}=\lim _{\Delta t\to 0}{\Delta x \over \Delta t}$

The instantaneous acceleration vector a of an object that has position at time t is given by x(t) is

$\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{x}}{dt^2}$

The equation for an object's velocity can be obtained mathematically by taking the integral of the equation for its acceleration beginning from some initial period time $t_0$ to some point in time later $t_n$ .

The final velocity v_f of an object which starts with velocity v_i and then accelerates at constant acceleration a for a period of time t is:

$v_f = v_i + at$

The average velocity of an object undergoing constant acceleration is (v_i + v_f)/2. To find the displacement d of such an accelerating object during a time interval t, substitute this expression into the first formula to get:

$d=t\times {\frac {(v_{i}+v_{f})}{2}}$

When only the object's initial velocity is known, the expression

$d=v_{i}t+{\frac {(at^{2})}{2}}$

can be used. These basic equations for final velocity and displacement can be combined to form an equation that is independent of time, also known as Torricelli's Equation:

$v_{f}^{2}=v_{i}^{2}+2ad$

The above equations are valid for both classical mechanics and special relativity. Where classical mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in classical mechanics, all observers agree on 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.

The kinetic energy (energy of motion) of a moving object is linear with both its mass and the square of its velocity:

$E_{v}={\begin{matrix}{\frac {1}{2}}\end{matrix}}mv^{2}$

The kinetic energy is a scalar quantity.

Polar coordinates

In polar coordinates, a two-dimensional velocity can be decomposed into a radial velocity, defined as the component of velocity away from or toward the origin, and transverse velocity, the component of velocity along a circle centred at the origin, and equal to the distance to the origin times the angular velocity.

Angular momentum in scalar form is the distance to the origin times the transverse speed, or equivalently, the distance squared times the angular speed, with a plus or minus to distinguish clockwise and anti-clockwise direction.

If forces are in the radial direction only, 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

• Terminal velocity