A sphere of mass m moves with a velocity 2v and collides inelastically with another identical sphere of mass m. After collision the first mass moves with velocity v in a direction perpendicular to the initial direction of motion . Find the speed of the second sphere after collision .

Options

A. v
B. `vsqrt(5)`
C. `2/sqrt3 v`
D. `v/sqrt3`

Answer

To solve the problem of two identical spheres colliding inelastically, we can use the principle of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Understand the Initial Conditions
- Sphere 1 (mass = m) is moving with a velocity of \(2v\) in the x-direction.
- Sphere 2 (mass = m) is initially at rest.

### Step 2: Set Up the Momentum Conservation Equations
Since the collision is inelastic, we will conserve momentum in both the x and y directions.

**Before the collision:**
- Total momentum in the x-direction: \(p_{x, initial} = 2mv + 0 = 2mv\)
- Total momentum in the y-direction: \(p_{y, initial} = 0\)

**After the collision:**
- Sphere 1 moves with a velocity \(v\) in the y-direction.
- Let Sphere 2 have a velocity \(v_x\) in the x-direction and \(v_y\) in the y-direction.

### Step 3: Write the Momentum Conservation Equations
1. **For the x-direction:**
\[ 2mv = mv_x + mv_y \]
Simplifying gives:
\[ 2v = v_x + v_y \quad (1) \]

2. **For the y-direction:**
\[ {0} = mv - mv_y \]
Simplifying gives:
\[ mv = mv_y \quad (2) \]
Therefore:
\[ v_y = v \quad (3) \]

### Step 4: Substitute \(v_y\) into the x-direction Equation
Substituting \(v_y = v\) from equation (3) into equation (1):
\[ 2v = v_x + v \]
Rearranging gives:
\[ v_x = 2v - v = v \quad (4) \]

### Step 5: Calculate the Speed of Sphere 2
Now, we have the components of the velocity of Sphere 2:
- \(v_x = v\)
- \(v_y = v\)

The speed \(v_2\) of Sphere 2 can be found using the Pythagorean theorem:
\[ v_2 = \sqrt{v_x^2 + v_y^2} \]
Substituting the values from (4):
\[ v_2 = \sqrt{(v)^2 + (v)^2} = \sqrt{2v^2} = v\sqrt{2} \]

### Final Answer
The speed of the second sphere after the collision is:
\[ v_2 = v\sqrt{2} \]

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