After a totally inelastic collision, two objects of the same mass and same initial speeds are found to move together at half of their initial speeds. The angle between the initial velocities of the objects is

Options

A. `30^(@)`
B. `60^(@)`
C. `45^(@)`
D. `37^(@)`

Explanation

To solve the problem, we need to apply the principle of conservation of momentum. Let's break it down step by step.

### Step 1: Understand the Situation
We have two objects of the same mass (let's call it \( m \)) and the same initial speed (let's call it \( v \)). After a totally inelastic collision, they move together with a speed of \( \frac{v}{2} \). We need to find the angle \( \theta \) between their initial velocities.

### Step 2: Write the Conservation of Momentum Equation
In a totally inelastic collision, the total momentum before the collision is equal to the total momentum after the collision.

Let the initial velocities of the two objects be \( \vec{v_1} \) and \( \vec{v_2} \). The momentum before the collision can be expressed as:
\[ \vec{P}_{\text{initial}} = m \vec{v_1} + m \vec{v_2} \]

After the collision, the two objects stick together and move with a common velocity \( \vec{v_f} = \frac{v}{2} \). The total momentum after the collision is:
\[ \vec{P}_{\text{final}} = 2m \cdot \frac{v}{2} = mv \]

### Step 3: Set Up the Momentum Equation
Now, we can write the equation for conservation of momentum:
\[ m \vec{v_1} + m \vec{v_2} = mv \]

Dividing through by \( m \) gives:
\[ \vec{v_1} + \vec{v_2} = v \]

### Step 4: Resolve the Velocities
Assuming \( \vec{v_1} \) is along the x-axis and \( \vec{v_2} \) makes an angle \( \theta \) with \( \vec{v_1} \), we can express the velocities in terms of their components:
- \( \vec{v_1} = v \hat{i} \)
- \( \vec{v_2} = v (\cos \theta \hat{i} + \sin \theta \hat{j}) \)

Now, substituting these into the momentum equation:
\[ v \hat{i} + v (\cos \theta \hat{i} + \sin \theta \hat{j}) = v \]

### Step 5: Simplify the Equation
This simplifies to:
\[ v (1 + \cos \theta) \hat{i} + v \sin \theta \hat{j} = v \]

Dividing through by \( v \) (assuming \( v \neq 0 \)):
\[ (1 + \cos \theta) \hat{i} + \sin \theta \hat{j} = 1 \]

### Step 6: Equate Components
From the x-component:
\[ 1 + \cos \theta = 1 \implies \cos \theta = 0 \]

From the y-component:
\[ \sin \theta = 0 \]

### Step 7: Find the Angle
The equation \( \cos \theta = 0 \) implies that \( \theta = 90^\circ \) or \( \theta = 270^\circ \). However, since we are looking for the angle between the two velocities before the collision, we need to find the angle between the two initial velocities.

Since both objects have the same speed and the angle between them is \( 2\theta \), we have:
\[ \text{Angle between initial velocities} = 2\theta = 120^\circ \]

### Final Answer
The angle between the initial velocities of the objects is \( 120^\circ \).

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