Question

Two persons of mass and are standing at the two ends A and B respectively, of a trolley of mass M as shown.

When the person standing at 'A' position who was initially at rest jumps from the trolley , towards left with $u_{rel}$ with respect to the trolley then :

• A. the trolley moves towards right
• B. the trolley rebounds with velocity $\frac{m_1{u}_{rel}}{m_1+m_2+M}$
• C. the centre of mass of the system remains stationary
• D. all the above

Answer 1

The correct option is D all the above

The center of mass is an unique point where the weighted relative mass is distributed which sums to be zero.

This is a point which cause force to be applied for linear acceleration without an angular acceleration.

Since no external force acts on the systeam , centre of mass of the system remains constant.

So centre of mass of the system remains stationary

so option C is correct

SInce centre of mass of system is sationary the trolly moves towards right .

so option A is correct

Let the velocity of trolley be $v_1$ towards and the velocity of person A be

$v_2$ towards left

Then $u_{(}rel)$ =$v_1+v_2$

Since initially they are in rest,So initial momentum is 0

Now final momentum :$(M+m_2)v_1-m_1\left(v_2\right)$

Now by conservation of momentum

$p\left(initial\right)=p\left(final\right)$

$(M+m_2)v_1-m_1\left(v_2\right)=0$

$(M+m_2)v_1=m_1\left(v_2\right)$

$v_2=\left[\right(M+m_2\left)/m_1\right]v_1$

Now u(rel) = $\left[\right(M+m_1+m_2\left)/m_1\right]v_1$

Therefore

the trolley rebounds with velocity $\frac{m_1{u}_{rel}}{m_1+m_2+M}$

Option B is also correct.

so all the above are correct