Question

A body moving with a uniform acceleration describes $12m$ in the third second of its motion and $20m$ in the fifth second. Find the distance covered in the eighth second.

Answer 1

$\text{Step 1: Expression for nth second displacement}$

Displacement in $n^{th}$ second is given by

$S_n=u+\frac{1}{2}a(2n-1)$

Here,

$u=$ initial velocity; $a=$ acceleration; $n=n^{th}$ second

$\text{Step 2: Distance covered in 3rd, 5th and 8th sec}$

The body is moving with uniform acceleration hence $a$ remains constant throughout the motion of the body.

Distance covered in $3^{rd}$ sec: $S_3=u+\frac{1}{2}a(2\times 3-1)$

$\Rightarrow$ $12=u+\frac{5}{2}a$ $....\left(1\right)$

Distance covered in $5^{th}$ sec: $S_5=u+\frac{1}{2}a(2\times 5-1)$

$\Rightarrow$ $20=u+\frac{9}{2}a$ $....\left(2\right)$

Distance covered in $8^{th}$ sec: $S_8=u+\frac{1}{2}a(2\times 8-1)$ $....\left(3\right)$

$\text{Step 3: Solving equations}$

Subtracting equation $\left(1\right)$ from equation $\left(2\right)$

$8=2a$

$\Rightarrow$ $a=4m/s^2$

From equation $\left(1\right)\Rightarrow$ $12=u+\frac{5}{2}\times 4$ $\Rightarrow u=2m/s$

Substitute the value of $u$ and $a$ in equation $\left(3\right)$

$S_8=2+\frac{1}{2}\times 4(2\times 8-1)$ $=32m$

Hence, the distance covered in 8th second is $32m$