A ball of mass `'m'` is thrown at an angle is `'theta'` with the horizontal with an initial velcoity 'u'. The change in its momentum during its flight in a time interval of 't' is .

Options

A. `"mgt"`
B. `"mgt" cos theta`
C. `"mgt sin" theta`
D. `½ "mgt"`

Explanation

To find the change in momentum of a ball thrown at an angle θ with the horizontal, we can follow these steps:

### Step 1: Understand the Initial Conditions
The ball has an initial mass \( m \) and is thrown with an initial velocity \( u \) at an angle \( \theta \).

### Step 2: Resolve the Initial Velocity
The initial velocity \( u \) can be resolved into horizontal and vertical components:
- Horizontal component: \( u_x = u \cos(\theta) \)
- Vertical component: \( u_y = u \sin(\theta) \)

### Step 3: Determine the Time of Flight
The time of flight \( t \) can be determined from the vertical motion. However, for the change in momentum, we will focus on the vertical component because the horizontal component remains constant (no horizontal forces acting).

### Step 4: Calculate the Change in Momentum
The change in momentum (Δp) can be calculated as follows:
- The momentum in the vertical direction before the ball is thrown is \( p_{initial} = m u_y = m (u \sin(\theta)) \).
- The momentum in the vertical direction at time \( t \) can be calculated considering the effect of gravity. The vertical velocity at time \( t \) is given by:
\[ v_y = u_y - g t = u \sin(\theta) - g t \]
- Therefore, the momentum at time \( t \) is:
\[ p_{final} = m v_y = m (u \sin(\theta) - g t) \]

### Step 5: Calculate the Change in Momentum
Now, we can find the change in momentum:
\[ \Delta p = p_{final} - p_{initial} \]
Substituting the values:
\[ \Delta p = m (u \sin(\theta) - g t) - m (u \sin(\theta)) \]
\[ \Delta p = -m g t \]

### Conclusion
The change in momentum during the flight of the ball in the time interval \( t \) is:
\[ \Delta p = -m g t \]