Question

A child will be standing at one end of a long trolley which is in motion with a speed $v$ on a smooth horizontal track. If the child begins running towards the other end of the trolley with a speed of $u$. What will be the speed with the centre of mass of the system which includes trolley and child will move?
$\begin{align}
  & A.zero \\
 & B.v+u \\
 & C.v \\
 & D.v-u \\
\end{align}$

Answer 1

Hint: The resultant external force with which the boy is moving will be zero. As we already know the external force can be found by taking the product of the mass of the system and the acceleration of the centre of mass of the system of bodies. This will help you in answering this question.

Complete step by step solution:
As we can see the speed with which a long trolley on the smooth horizontal track is moving can be shown as $v$. The child begins running towards the other end of the trolley with a speed mentioned in the question as $u$.
Therefore the resultant external force with which the boy is moving will be zero. This can be written as an equation given as,
${{F}_{ext}}=0N$
As we already know the external force can be found by taking the product of the mass of the system and the acceleration of the centre of mass of the system of bodies. This can be written as an equation given as,
${{F}_{ext}}=m{{\vec{a}}_{com}}$
Where $m$ be the mass of the system and ${{\vec{a}}_{com}}$be the acceleration of the centre of mass of the system. As the external force is zero we can write that,
$m{{\vec{a}}_{com}}=0$
That is the acceleration of the centre of mass will be zero.
$\therefore {{\vec{a}}_{com}}=0$
Hence the centre of mass of the system will not be moving. Therefore the velocity of the system will be a constant. That is, the velocity of the centre of mass is found to be $v$.

So, the correct answer is “Option C”.

Note: The centre of mass of the body can be defined as the imaginary point in the body or outside the body where the whole mass of the system has been considered to be concentrated. Every property of the system will be shown by this centre of mass. This is basically for the simplification of the calculations in physics.