Question

Derive the following equation for a uniformly accelerated motion:

• $S=ut+\frac{1}{2}a{t}^2$

Answer 1

Consider the linear motion of a body with an initial velocity 𝑢. It accelerates uniformly and in time 𝑡, it acquires the final velocity 𝑣. Velocity-time graph of the body will be a straight line 𝐴𝐵 as shown in the graph. The body covers a distance S in time t.
From the graph,
Initial velocity (at 𝑡 = 0) = 𝑂𝐴 = 𝑢
Final velocity (at time 𝑡) = 𝑂𝐸 = v
We know that the area under velocity-time graph will give the total distance travelled by the body.
So, distance S travelled in time t
$=$ Area of trapezium OABC
= Area of rectangle OADC +
Area of triangle ABD
$S=(OA\times OC)+\{\frac{1}{2}\times (AD\times BD\left)\right\}=(u\times t)+\{\frac{1}{2}\times t\times (v-u\left)\right\}$
From equation $v=u+at$, substituting the value of $\text{v}-\text{u}=$ at in the above equation;
$S=(u\times t)+\{\frac{1}{2}\times t\times at\}S=ut+\frac{1}{2}a{t}^2$