Question

A boy of mass 50kg running at 5m/s jumps on to a 20kg trolley travelling in the same direction at $1.5m/s$ .Find their common velocity.

Answer 1

Here, both trolley and boy are travelling with two different velocities. When the boy jumps on the trolley, they both become a system and start travelling with the same velocity. Here, according to the law of conservation of momentum their initial and final momentum should be equal. Hence, by finding the initial and final momentum of the boy and trolley and equating them will give us their common velocity.
Formula used:
$\text{Initial momentum = final momentum}$
$\text{Momentum = mv}$

Complete answer:
According to conservation of momentum, the initial and final momentum should be equal.
i.e.,
$\text{Initial momentum = final momentum}$ ----- (1)
Given,
Mass of boy, ${{m}_{1}}=50kg$
Velocity of boy, ${{v}_{1}}=5m/s$
Mass of trolley, ${{m}_{2}}=20kg$
Velocity of trolley, ${{v}_{2}}=1.5m/s$
Then,
Initial momentum of boy and trolley,
${{p}_{initial}}={{m}_{1}}{{v}_{1}}+{{m}_{2}}{{v}_{2}}=50\times 5+20\times 1.5=280kgm/s$
When the boy jumps on the trolley, the both will have the same velocity.
Final momentum of boy and trolley,
${{p}_{final}}=\left( {{m}_{1}}+{{m}_{2}} \right)v=\left( 50+20 \right)v=70v$
Where, $v$is the common velocity.
Then, from equation 1,
${{\text{p}}_{initial}}\text{ = }{{\text{p}}_{final}}$
By substituting the values, we get,
$280=70v$
$v=\dfrac{280}{70}=4m/s$
Common velocity of boy and trolley is $4m/s$

Note:
According to conservation of momentum, momentum never changes in an isolated system, that is, the total momentum of the system remains constant. Momentum is equal to the mass of a particle multiplied by its velocity and is equivalent to the force needed to bring the object to stop in a unit length of time. For a collection of several objects, the total momentum is the sum of their individual momentum. Momentum is a vector quantity, involving both the magnitude and direction of motion, so that the momentum of objects going in opposite directions can cancel and result in an overall sum of zero.