Question

Derive the second equation of motion by a method other than the graphical method.

Answer 1

Step 1: Second Equation of motion

1. Equations of motion are applicable only when a particle moves with constant acceleration.
2. Let ‘u’ be the initial velocity at time t = 0, ‘v’ is the final velocity at time t, ‘s’ is the displacement done by the particle, and ‘a' be the constant acceleration in the motion of the particle.
3. The second equation of motion is $s=ut+\frac{1}{2}a{t}^2$

Step 2: Derivation of the second equation

1. Acceleration is defined as the rate of change of velocity of the particle.
2. So, $acceleration=\frac{finalvelocity-initialvelocity}{finaltime-intialtime}$= $\frac{v-u}{t-0}$
3. So, $a=\frac{v-u}{t}$$\Rightarrow v-u=at$ $\Rightarrow v=u+at$ -----(1)
4. As the motion is uniformly accelerated, the value of average velocity = $\frac{initialvelocity+finalvelocity}{2}$
5. So we can write $V_{avg}=\frac{u+v}{2}$
6. Now we can calculate the displacement of the particle by multiplying the average velocity of the particle by the time taken by the particle.

Displacement (s) = average velocity (V _avg)$\times$time(t)

$\Rightarrow$$s=\frac{\left(u+v\right)}{2}\times t$

Now put the value of v from equation (1) in this expression, we get

$\Rightarrow$$s=\frac{\left(u+u+at\right)}{2}\times t$

$\Rightarrow$$s=\frac{1}{2}\times 2u\times t+\frac{1}{2}\times at\times t$

$\Rightarrow$$s=ut+\frac{1}{2}a{t}^2$

Thus the second equation of motion ( $s=ut+\frac{1}{2}a{t}^2$) can be derived.