Q.Two billiard balls each of mass $0.05\ \text{kg}$ moving in opposite directions with speed $6\ \text{m s}^{-1}$ collide and rebound with the same speed. What is the impulse imparted to each ball due to the other?

The key idea here is the Impulse-Momentum Theorem, which states that the impulse imparted to an object equals its change in momentum.

Consider one of the billiard balls. Let its initial direction of motion be positive.

1. The initial momentum of the ball is $p_i = m u$.
2. After collision, the ball rebounds with the same speed in the opposite direction. So, its final velocity is $-u$, and its final momentum is $p_f = m (-u)$.
3. The impulse $\vec{J}$ imparted to this ball is the change in its momentum: $\vec{J} = p_f - p_i = m(-u) - mu = -2mu$.
4. Substituting the given values, $m = 0.05\ \text{kg}$ and $u = 6\ \text{m s}^{-1}$: …

When objects collide, the forces involved are often very large and act for a very short duration. It's usually difficult, if not impossible, to measure these forces directly or the exact time interval of the collision. This is where the concept of impulse becomes incredibly useful.

Impulse ($\vec{J}$) is a measure of the overall effect of a force acting over a period of time. It is defined as the integral of force with respect to time: $\vec{J} = \int \vec{F} \, dt$. The Impulse-Momentum Theorem provides a powerful shortcut, stating that the impulse imparted to an object is equal to the change in its linear momentum ($\Delta \vec{p}$).

This theorem is particularly helpful in collision problems because we often know the initial and final velocities of the objects, even if we don't know the details of the forces during the collision. The key is to remember that momentum and impulse are vector quantities, meaning their direction is just as important as their magnitude.

Let's apply this to the billiard balls.

1. Identify Given Information and Set Up Coordinate System:

   We are given:

   • Mass of each ball, $m = 0.05\ \text{kg}$.
   • Initial speed of each ball, $u = 6\ \text{m s}^{-1}$.
   • Final speed of each ball, $v = 6\ \text{m s}^{-1}$ (same speed).

   Since the balls are moving in opposite directions and rebound with the same speed, their directions of motion reverse. Let's define a coordinate system: we'll consider motion to the right as positive ($+\hat{i}$) and motion to the left as negative ($-\hat{i}$).

2. Determine Initial and Final Velocities for Ball 1:

   Let's consider the first ball (Ball 1).

   • Its initial velocity is $\vec{u}_1 = +6\ \hat{i}\ \text{m s}^{-1}$ (moving to the right).
   • After collision, it rebounds, meaning its direction reverses. So, its final velocity is $\vec{v}_1 = -6\ \hat{i}\ \text{m s}^{-1}$ (moving to the left).

3. Calculate Change in Momentum for Ball 1:

   The change in momentum for Ball 1 is $\Delta \vec{p}_1 = m\vec{v}_1 - m\vec{u}_1$.

> [!WARNING]
> A common mistake is to treat speed as velocity and simply subtract magnitudes. Always use vector velocities (with appropriate signs) when calculating change in momentum. If you just subtracted speeds, you would get $m(6-6)=0$, which is incorrect. The change in velocity is $v - (-u)$ or $-v - u$, depending on your initial direction choice.

4. Apply Impulse-Momentum Theorem for Ball 1:

According to the Impulse-Momentum Theorem, the impulse imparted to Ball 1 is equal to its change in momentum:

This means the impulse on Ball 1 has a magnitude of $0.6\ \text{kg m s}^{-1}$ and is directed to the left (opposite to its initial motion).

5. Determine Initial and Final Velocities for Ball 2:

Now consider the second ball (Ball 2). …

Concept: Impulse–Momentum Theorem in a Collision

Step 1: Set up the coordinate axis

Let motion to the right be $+\hat i$. Ball 1: $u_1=+6\hat i\ \text{m s}^{-1}$; Ball 2:

$u_2=-6\hat i\ \text{m s}^{-1}$. Both rebound with the same speed, so their directions

reverse: $v_1=-6\hat i\ \text{m s}^{-1}$, $v_2=+6\hat i\ \text{m s}^{-1}$.

Step 2: Impulse on Ball 1

Step 3: Impulse on Ball 2

$\vec J_2=m(\vec v_2-\vec u_2)=0.05(6-(-6))\hat i=+0.6\hat i\ \text{kg m s}^{-1}$ …