Question

A ball of mass $m_1=4\text{g}$ thrown towards a wall with a velocity $u_1=4\text{m/s}$ collides against the wall elastically as shown in figure. What will the velocity of the ball after collision?

• A. $2\text{m∕s opposite to the original direction}$
• B. $4\text{m∕s opposite to the original direction}$
• C. $8\text{m∕s opposite to the original direction}$
• D. $6\text{m∕s opposite to the original direction}$

Answer 1

The correct option is B $4\text{m∕s opposite to the original direction}$

Given:
Mass of ball; $m_1=4\text{g}=4\times {10}^{-3}\text{kg}$
Mass of wall; $m_2=?$
Initial velocity of ball; $u_1=4\text{m/s}$
Initial velocity of wall; $u_2=0\text{m/s}$
Final velocity of wall; $v_2=0\text{m/s}$
Let final velocity of ball be $v_1$.

In elastic one dimensional collision, relative velocity before and after collision is equal.
$\therefore u_1-u_2=v_2-v_1$
$\Rightarrow 4-0=0-v_1$
$\Rightarrow v_1=-4\text{m/s}$
Here $-ve$ indicates that the ball will have a final velocity of $4\text{m/s}$ opposite to its original direction.

ALTERNATE SOLUTION

We can solve this problem with the concept of collision with heavy mass object. Assume wall as a heavy mass $m_2$ such that $m_2>>.m_1$

From the concept of collision we know
$v_1=\left(\frac{m_1-m_2}{m_1+m_2}\right)u_1+\frac{2m_2{u}_2}{m_1+m_2}$
$[\text{As}{m}_2>>>m_1]$
$\Rightarrow v_1=-u_1+2u_2=(-4+2\times 0)\text{m/s}=-4\text{m/s}$
Here $-ve$ indicates ball will have final velocity of $4\text{m/s}$ opposite to its original direction.