Q.(a) When a molecule (or an elastic ball) hits a (massive) wall, it rebounds with the same speed. When a ball hits a massive bat held firmly, the same thing happens. However, when the bat is moving towards the ball, the ball rebounds with a different speed. Does the ball move faster or slower? (Ch. 5 will refresh your memory on elastic collisions.)

1. Key idea: Relative velocity in an elastic collision. When a massive wall is stationary, the ball rebounds with speed $v$. If the wall moves toward the ball with speed $u$, the relative speed of approach is $v+u$. In an elastic collision with a much heavier object, the relative speed of separation equals the relative speed of approach. So the ball’s rebound speed relative to the ground becomes $v+2u$ — faster.
2. Kinetic theory explanation: A piston moving inward is like a moving wall. Gas molecules hitting the advancing piston rebound with increased speed (as in part a). This raises the average kinetic energy of the molecules. Since temperature is proportional to average kinetic energy, the gas temperature rises.
3. When the compressed gas pushes the piston outward, the piston moves away from the molecules. Now molecules hit a receding wall and rebound with reduced speed. Their average kinetic energy drops, so the gas cools. You would observe a drop in temperature of the gas. …

(a) Ball hitting a moving bat — faster or slower?

Think of a ball approaching a wall at speed $v$. If the wall is stationary, the ball rebounds with speed $v$ — elastic collision, kinetic energy conserved. Now imagine the wall (or bat) moves towards the ball with speed $u$. From the ball’s perspective, the wall is coming at it faster. In the frame of the bat, the ball approaches at $v + u$ and leaves at $v + u$ (elastic). Transforming back to the ground frame, the ball’s rebound speed becomes $v + 2u$. So the ball leaves faster than it came.

Conversely, if the bat moves away from the ball, the rebound speed is smaller. So: moving towards → faster; moving away → slower.

(b) Compressing a gas: why temperature rises

In kinetic theory, gas molecules are tiny elastic balls. The piston is like a moving wall. When you push the piston in (compression), it moves towards the molecules. Each time a molecule hits the advancing piston, it rebounds with a higher speed — just like the ball in part (a). This increases the molecule’s kinetic energy.

Since temperature is proportional to the average kinetic energy of molecules ($\frac{3}{2}kT = \frac{1}{2}m\langle v^2 \rangle$), the gas temperature rises.

(c) Expanding gas: what happens

When the compressed gas pushes the piston out (expansion), the piston moves away from the molecules. Now a molecule hitting the retreating piston rebounds with a lower speed — the opposite of part (a). Each collision reduces the molecule’s kinetic energy. So the average kinetic energy drops, and the gas cools down.

You would observe the gas temperature decreasing. If the expansion is rapid, the cooling can be dramatic — you might even see frost form on the cylinder.

(d) Why a heavy bat helps Sachin Tendulkar …

The unifying picture for all four parts is momentum transfer in a moving frame: a stationary wall/bat/piston just reverses a particle's velocity, but one moving towards the particle adds extra closing speed that the particle carries away on rebound — the same elastic-collision-with-a-moving-object idea from Ch. 5. Applying it once: an advancing piston (compression) speeds molecules up $\to$ higher KE $\to$ higher $T$; a retreating piston (expansion) slows them down $\to$ cools the gas; and a heavy bat acts like a near-stationary wall, so from $v_{r …