Animated Solution for Physics - System of Particles: A man (mass = $50\text{ kg}$) and his son (mass = $20\text{ kg}$) are standing on a frictionless surface facing each other. The man pushes his son, so that he starts moving at a speed of $0.70{\text{ ms}}^{-1}$ with respect to the man. The speed of the man with respect to the surface is

Imagine you are watching a man and his son standing completely still on a perfectly smooth, frictionless ice rink. They are facing each other. The man has a mass of $50\text{ kg}$, and the son has a mass of $20\text{ kg}$. Right now, nothing is moving. The initial velocities are zero.

Why Momentum is Conserved

Now, the man pushes his son. Because the surface is frictionless, there is absolutely no external horizontal force acting on this two-person system. What does that mean? It means we can apply the Principle of Conservation of Linear Momentum!

The total momentum before the push was zero, so the total momentum after the push must also be zero. If the man moves backwards with speed $v_1$, and the son moves forwards with speed $v_2$, their momenta must be equal and opposite.

Setting up the Equations

Let's plug in the numbers we know. The man's mass is $50\text{ kg}$, and the son's mass is $20\text{ kg}$.

If we rearrange this to solve for the son's speed, $v_2$, we get $50$ divided by $20$ times $v_1$. This simplifies beautifully to:

The son is lighter, so he moves $2.5$ times faster than his dad!

The problem gives us a crucial piece of information: the relative speed of the son with respect to the man is $0.70\text{ m/s}$. Notice how they are moving away from each other in opposite directions.

When two objects move in opposite directions, their relative speed is simply the sum of their individual speeds. So, the relative velocity, $v_{\text{rel}}$, is equal to $v_1+v_2$.

We are almost there! Let's substitute the expression we found for $v_2$ into our relative velocity equation. We replace $v_2$ with $2.5v_1$.

Adding those up, we get:

To find $v_1$, the speed of the man, we just divide $0.70$ by $3.5$.

This is the speed of the man relative to the stationary ice surface. The correct option is (b).

Where did the energy come from?

What if the question asked for the son's speed? You could easily find it by multiplying this by $2.5$, giving $0.50\text{ m/s}$.

Also, think about the energy. Initially, the kinetic energy of the system was zero. Finally, both the man and the son have positive kinetic energy. Where did this energy come from? It didn't magically appear; it came from the internal chemical energy of the man's muscles doing work during the push! Keep thinking about these deeper concepts, as they often form the basis of more advanced JEE questions.