# Define an expression for distance travelled by a body with uniform acceleration. Acceleration is the measure of velocity change, mathematically it is expressed as  `a(t) = v'(t),` where `t` is time, `v` is the velocity and `a` is the acceleration. Note that velocity is a vector quantity and therefore acceleration is a vector, too. Velocity, in turn, is the derivative of displacement `d`...

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Acceleration is the measure of velocity change, mathematically it is expressed as  `a(t) = v'(t),` where `t` is time, `v` is the velocity and `a` is the acceleration. Note that velocity is a vector quantity and therefore acceleration is a vector, too. Velocity, in turn, is the derivative of displacement `d` (also a vector quantity).

If the acceleration of a body is a constant vector, its velocity and displacement are collinear with the acceleration, so the movement is along a straight line. This line is suitable for projection as an axis. Denote the magnitude of the acceleration as `a,` the speed at `t = 0` as `v_0` and let the starting position to be zero.

Integrating the equality `v'(t) = a` we obtain `v(t) = at + v_0,` integrating `d'(t) = v(t) = at + v_0` we obtain `d(t) = (a t^2)/2 + v_0 t.` This is the distance travelled since `t=0.` The distance travelled between moments `t_1` and `t_2` is

`a(t_2^2- t_1^2)/2 + v_0 (t_2 - t_1).`