A particle of mass `m` moving with a speed `v` hits elastically another staionary particle of mass `2m` on a smooth horizontal circular tube of radius `r`. Find the time when the next collision will take place?

Explanation

To solve the problem step-by-step, we will analyze the elastic collision between the two particles and determine the time until their next collision.

### Step 1: Understand the Initial Conditions
- We have two particles:
- Particle 1 (mass = m) moving with speed v.
- Particle 2 (mass = 2m) is stationary.
- The collision occurs on a smooth horizontal circular tube of radius r.

### Step 2: Apply Conservation of Momentum
Before the collision, the total momentum of the system is:
\[ p_{\text{initial}} = mv + 0 = mv \]

After the collision, let:
- \( v_1 \) be the velocity of particle 1 (mass m) after the collision.
- \( v_2 \) be the velocity of particle 2 (mass 2m) after the collision.

Using the conservation of momentum:
\[ mv = mv_1 + 2mv_2 \]
This simplifies to:
\[ v = v_1 + 2v_2 \quad (1) \]

### Step 3: Apply the Coefficient of Restitution
For elastic collisions, the coefficient of restitution \( e = 1 \). The equation for the coefficient of restitution is:
\[ e = \frac{\text{Relative velocity after collision}}{\text{Relative velocity before collision}} \]

Before the collision, the relative velocity of approach is:
\[ v_{\text{approach}} = v - 0 = v \]

After the collision, the relative velocity of separation is:
\[ v_{\text{separation}} = v_2 - v_1 \]

Setting up the equation:
\[ 1 = \frac{v_2 - v_1}{v} \]
This leads to:
\[ v_2 - v_1 = v \quad (2) \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( v = v_1 + 2v_2 \) (from momentum conservation)
2. \( v_2 - v_1 = v \) (from restitution)

From equation (2), we can express \( v_1 \) in terms of \( v_2 \):
\[ v_1 = v_2 - v \]

Substituting this into equation (1):
\[ v = (v_2 - v) + 2v_2 \]
This simplifies to:
\[ v = 3v_2 - v \]
\[ 2v = 3v_2 \]
Thus,
\[ v_2 = \frac{2}{3}v \]

Now substituting \( v_2 \) back into equation (2) to find \( v_1 \):
\[ v_1 = \frac{2}{3}v - v = -\frac{1}{3}v \]

### Step 5: Determine the Time Until Next Collision
After the collision, both particles will move in opposite directions along the circular tube. The distance they need to cover to collide again is the circumference of the tube:
\[ \text{Circumference} = 2\pi r \]

The relative velocity between the two particles after the collision is:
\[ v_{\text{relative}} = v_1 + v_2 = -\frac{1}{3}v + \frac{2}{3}v = \frac{1}{3}v \]

The time \( t \) until the next collision is given by:
\[ t = \frac{\text{Distance}}{\text{Relative Velocity}} = \frac{2\pi r}{\frac{1}{3}v} = \frac{6\pi r}{v} \]

### Final Answer
The time until the next collision will take place is:
\[ t = \frac{6\pi r}{v} \]

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