# Derive the distance function for the motion of an accelerated body.

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Tushar Chandra | Certified Educator

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When a body with initial velocity u is accelerated at a rate a for t seconds the total distance traveled by the body is s = u*t + (1/2)*a*t^2.

This can be derived using integration. Use the basic formula that relates, distance time and speed, the distance traveled by an object is the product of speed and time.

In an instant dt, the distance moved by the object is v*dt. The acceleration of a body is the rate of change of its speed, a = (v - u)/t. At any time t, the speed of a body is equal to v = u + a*t

Take the integral `int v*dt `

`s = int v*dt`

= `int u + a*t dt`

= `u*t + (1/2)a*t^2`

The formula for distance traveled by a body accelerating at a rate a and with initial velocty u in time t is `s = u*t + (1/2)*a*t^2`

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