Question

derive the second equation of motion s =ut+1/2at2 graphically

Answer 1

Plotting time along the horizontal axis and velocity along the vertical axis in a graph and considering “u” as the velocity of a particle moving with uniform acceleration “a” and after a given period of time “t” let the velocity be “v” and the distance covered is “s”.

The average velocity = (v + u)/2 …………………………….. Equation 1

Distance(S) travelled in “t” seconds = {(v + u)/2} X t

 Again, v = u + at………….Equation 2

 Using both the equations we get, S = [(u +at +u)/2] x T

On simplification we get S = ut + ½ at2.

Answer 2

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Consider the linear motion of a body with an initial velocity u. Let the body accelerate uniformly and acquire a final velocity v after time t. The velocity–time graph is a straight line AB as shown below.

At t = 0, initial velocity = u = OA

At t = t, final velocity = v = OC

The distance S travelled in time t = area of the trapezium OABD

s = (1/2) x (OA + DB) × OD

s = (1/2) x (u + v) × t

Since v = u + at,

s = (1/2) x (u + u + at) × t

s = ut + (1/2) at2