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 vi to vf during a time interval t is given by:
- a = (vf - vi)/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 vf of an object which starts with velocity vi and then accelerates at constant acceleration a for a period of time t is:
- vf = vi + at
The average velocity of an object undergoing constant acceleration is (vf + vi)/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(vf + vi)/2
When only the object's initial velocity is known, the expression
- d = vit + (a't2)/2
can be used. These basic equations for final velocity and displacement can be combined to form an equation that is independent of time:
- vf2 = vi2 + 2ad
These simple equations become more complicated as velocities approach the speed of light, where the effects of special relativity start to become significant.