A particle of mass $m_1$ , making a head on elastic collision with another stationary ball of mass $m_2$ . If $m_1$ rebounds with half of its original speed then $\frac{m_1}{m_2}$ is

Collisions

NEET

$\frac{2}{3}$

$\frac{3}{2}$

$\frac{1}{3}$

Solution:

Let the initial velocity of mass $m_1$ be $u_1$ and the initial velocity of mass $m_2$ be $u_2 = 0$ (since it's stationary).

After the head-on elastic collision, mass $m_1$ rebounds with half of its original speed. So, its final velocity $v_1 = -\frac{u_1}{2}$ (negative sign indicates rebounding, i.e., opposite direction).

For a head-on elastic collision, the final velocity of the first mass ( $m_1$ ) is given by the formula:

$v_1 = \frac{(m_1 - m_2)u_1 + 2m_2 u_2}{m_1 + m_2}$

Substitute $u_2 = 0$ into the equation:

$v_1 = \frac{(m_1 - m_2)u_1}{m_1 + m_2}$

Now, substitute the given value of $v_1 = -\frac{u_1}{2}$ :

$-\frac{u_1}{2} = \frac{(m_1 - m_2)u_1}{m_1 + m_2}$

Assuming $u_1 \neq 0$ , we can cancel $u_1$ from both sides:

$-\frac{1}{2} = \frac{m_1 - m_2}{m_1 + m_2}$

Cross-multiply:

$-(m_1 + m_2) = 2(m_1 - m_2)$

$-m_1 - m_2 = 2m_1 - 2m_2$

Rearrange the terms to group $m_1$ and $m_2$ :

$2m_2 - m_2 = 2m_1 + m_1$

$m_2 = 3m_1$

To find the ratio $\frac{m_1}{m_2}$ :

$\frac{m_1}{m_2} = \frac{1}{3}$

Thus, the correct option is (3).

A particle of mass m1m_1m1​, making a head on elastic collision with another stationary ball of mass m2m_2m2​. If m1m_1m1​ rebounds with half of its original speed then m1m2\frac{m_1}{m_2}m2​m1​​ is

Solution:

Related Questions

A particle of mass $m_1$ , making a head on elastic collision with another stationary ball of mass $m_2$ . If $m_1$ rebounds with half of its original speed then $\frac{m_1}{m_2}$ is