A ball is thrown from a point with a speed 'v^(0)' at an elevation angle of `theta` . From the same point and at the same instant , a person starts running with a constant speed `('v_(0)')/(2) ` to catch the ball . Will the person be able to catch the ball ? If yes, what should be the angle of projection `theta` ?

Options

A. `Yes, 60^@`
B. `Yes, 30^@`
C. No
D. `Yes, 45^@`

Explanation

To determine whether the person can catch the ball and at what angle of projection \( \theta \) this can happen, we can analyze the motion of both the ball and the person.

### Step-by-Step Solution:

1. **Understand the Motion of the Ball**:
The ball is thrown with an initial speed \( v^0 \) at an angle \( \theta \). The horizontal and vertical components of the ball's velocity can be expressed as:
- Horizontal component: \( v_{x, \text{ball}} = v^0 \cos \theta \)
- Vertical component: \( v_{y, \text{ball}} = v^0 \sin \theta \)

2. **Determine the Horizontal Motion of the Person**:
The person starts running with a constant speed of \( \frac{v^0}{2} \). The horizontal displacement of the person after time \( T \) is:
\[ x_{\text{person}} = \left(\frac{v^0}{2}\right) T \]

3. **Calculate the Horizontal Displacement of the Ball**:
The horizontal displacement of the ball after time \( T \) is:
\[ x_{\text{ball}} = (v^0 \cos \theta) T \]

4. **Set the Horizontal Displacements Equal**:
For the person to catch the ball, their horizontal displacements must be equal at time \( T \):
\[ x_{\text{person}} = x_{\text{ball}} \]
Substituting the expressions we derived:
\[ \left(\frac{v^0}{2}\right) T = (v^0 \cos \theta) T \]

5. **Simplify the Equation**:
We can cancel \( T \) from both sides (assuming \( T \neq 0 \)):
\[ \frac{v^0}{2} = v^0 \cos \theta \]
Dividing both sides by \( v^0 \) (assuming \( v^0 \neq 0 \)):
\[ \frac{1}{2} = \cos \theta \]

6. **Find the Angle \( \theta \)**:
To find \( \theta \), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) \]
This gives us:
\[ \theta = 60^\circ \]

### Conclusion:
Yes, the person will be able to catch the ball if the angle of projection \( \theta \) is \( 60^\circ \).

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