Velocity

A vector measurement of the rate and direction of motion. The scalar absolute value of velocity is speed.

In Classical mechanics (that is, when the velocities involved are significantly less than the speed of light) the average speed v of an object moving a distance d during a time interval t is described by the simple formula:

v = d/t.

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

v = dx/dt.

Acceleration is the 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 = (v_f - v_i)/t.

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

a = d²x/(dt)²

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_f + v_i)/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(v_f + v_i)/2

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

d = v_it + (at²)/2

can be used. These basic equations for final velocity and displacement can be combined to form an equation that is independent of time:

v_f² = v_i² + 2ad

These simple equations become more complicated as velocities approach the speed of light, where the effects of special relativity start to become significant.

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