Two balls of equal masses moving with equal speed in mutually perpendicular directions, stick together after collision. If the balls were initially moving with a speed of `30/√2` ms^(-1) each, the speed of their combined mass after collision is

Options

A. 30 `ms^(-1)`
B. `15 ms^(-1)`
C. `30/√2 ms^(-1)`
D. `30sqrt2 ms^(-1)`

Answer

To solve the problem step by step, we will use the principles of conservation of momentum and the geometry of the situation.

### Step 1: Understand the scenario
We have two balls of equal mass moving with equal speed in mutually perpendicular directions. This means one ball is moving along the x-axis and the other along the y-axis.

### Step 2: Identify the initial speeds
Both balls are moving with a speed of \( \frac{30}{\sqrt{2}} \) m/s.

### Step 3: Set up the momentum conservation equations
Since the balls stick together after the collision, we can use the conservation of momentum. The momentum before the collision in the x-direction and y-direction must equal the momentum after the collision.

### Step 4: Calculate the momentum components
Let \( m \) be the mass of each ball and \( u = \frac{30}{\sqrt{2}} \) m/s be the speed of each ball.

- Momentum of ball 1 (moving along x-axis):
\[ p_x = m \cdot u = m \cdot \frac{30}{\sqrt{2}} \]

- Momentum of ball 2 (moving along y-axis):
\[ p_y = m \cdot u = m \cdot \frac{30}{\sqrt{2}} \]

### Step 5: Combine the momenta
After the collision, the two balls stick together, and their combined mass is \( 2m \). Let \( V \) be the speed of the combined mass after the collision.

Using the Pythagorean theorem, since the balls are moving at right angles to each other, the total momentum after the collision can be expressed as:
\[ \sqrt{(p_x)^2 + (p_y)^2} = 2mV \]

### Step 6: Substitute the values
Substituting the values of \( p_x \) and \( p_y \):
\[ \sqrt{\left(m \cdot \frac{30}{\sqrt{2}}\right)^2 + \left(m \cdot \frac{30}{\sqrt{2}}\right)^2} = 2mV \]
\[ \sqrt{2 \left(m \cdot \frac{30}{\sqrt{2}}\right)^2} = 2mV \]
\[ \sqrt{2} \cdot m \cdot \frac{30}{\sqrt{2}} = 2mV \]

### Step 7: Simplify the equation
Cancelling \( m \) from both sides (assuming \( m \neq 0 \)):
\[ 30 = 2V \]

### Step 8: Solve for \( V \)
Dividing both sides by 2:
\[ V = \frac{30}{2} = 15 \text{ m/s} \]

### Final Answer
The speed of the combined mass after the collision is \( 15 \) m/s.

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