A cricket fielder can throw the cricket ball with a speed `v_0`. If the throws the ball while running with speed u at an angle `theta` to the horizontal, 

Options

A. the effective angle to the horizontal at which the ball is projected in air as seen by a spectator is `tan^(-1)((v_0 sin theta)/(v_0 cos theta + u))`
B. time of flight is `(2v_0 sin theta)/(g)`
C. the distance (horizontal range) from the point of projection at which ar ball will land
`R = (2v_0 sin theta(v_0 cos theta +u))/g`
D. all the above are true.

Answer

To solve the problem, we need to analyze the motion of the cricket ball thrown by the fielder while running. We will break down the velocity components and calculate the time of flight and horizontal range.

### Step-by-step Solution:

1. **Identify the velocities involved**:
- The initial speed of the ball when thrown is \( v_0 \).
- The speed of the fielder running is \( u \).
- The angle of projection with respect to the horizontal is \( \theta \).

2. **Resolve the initial velocity into components**:
- The horizontal component of the ball's velocity when thrown is:
\[ v_{0x} = v_0 \cos(\theta) \]
- The vertical component of the ball's velocity when thrown is:
\[ v_{0y} = v_0 \sin(\theta) \]

3. **Determine the effective horizontal velocity**:
- The effective horizontal velocity when the ball is thrown while running is the sum of the horizontal component of the throw and the fielder's running speed:
\[ u_x = u + v_{0x} = u + v_0 \cos(\theta) \]

4. **Calculate the time of flight**:
- The time of flight \( T \) for a projectile is given by the formula:
\[ T = \frac{2 v_{0y}}{g} = \frac{2 v_0 \sin(\theta)}{g} \]
- Here, \( g \) is the acceleration due to gravity.

5. **Calculate the horizontal range**:
- The horizontal range \( R \) can be calculated using the effective horizontal velocity and the time of flight:
\[ R = u_x \cdot T = (u + v_0 \cos(\theta)) \cdot \frac{2 v_0 \sin(\theta)}{g} \]
- Substituting for \( u_x \):
\[ R = (u + v_0 \cos(\theta)) \cdot \frac{2 v_0 \sin(\theta)}{g} \]

6. **Summarize the results**:
- The statements regarding the projectile motion are true:
- The effective angle of projection can be calculated using the components.
- The time of flight is correctly derived.
- The horizontal range formula is valid.

### Conclusion:
All statements regarding the projectile motion of the cricket ball thrown by the fielder while running are true.