Question

Derive the second equation of motion.

Answer 1

Step 1: Distance

1. We know that the distance travelled for a moving object is simply the measure of the whole path travelled by the particle
   $\mathrm{Distance}=\mathrm{Average}\mathrm{velocity}\times \mathrm{time}$

Step 2: Deriving the second equation of motion

Since average velocity is given by the formula
$Averagevelocity=\frac{(u+v)}{2}$

Here, u is the initial velocity and v is the final velocity.

Now substituting the values of average velocity in the distance formula, we get
$s=\frac{(u+v)}{2}\times t$

Here, s is the distance and t is the time.

According to the first equation of motion,
$v=u+at$
Substituting the value of v in the above equation

$$\begin{gathered} s=\frac{(u+u+at)}{2}\times t \\ \Rightarrow \frac{(2u+at)}{2}\times t \end{gathered}$$
$s=\frac{(2ut+at²)}{2}$
$s=\frac{2ut}{2}+\frac{at²}{2}$
After solving the above equation we get $s=ut+\frac{1}{2}a{t}^2$
Hence the second equation of motion is $s=ut+\frac{1}{2}a{t}^2$.