A body of mass m is projected with intial speed u at an angle `theta` with the horizontal. The change in momentum of body after time t is :-

Options

A. m u `sin theta`
B. 2 m u `sin theta`
C. m g t
D. zero

Answer

To solve the problem of finding the change in momentum of a body of mass \( m \) projected with an initial speed \( u \) at an angle \( \theta \) with the horizontal after a time \( t \), we can follow these steps:

### Step 1: Determine the initial velocities in the x and y directions
The initial velocity \( u \) can be resolved into two components:
- Horizontal component (x-direction):
\[ u_x = u \cos \theta \]
- Vertical component (y-direction):
\[ u_y = u \sin \theta \]

### Step 2: Analyze the motion in the x-direction
In the x-direction, there is no acceleration (assuming no air resistance). Therefore, the horizontal velocity remains constant:
\[ v_x = u_x = u \cos \theta \]
Thus, the momentum in the x-direction does not change.

### Step 3: Analyze the motion in the y-direction
In the y-direction, the body is subject to gravitational acceleration \( g \) acting downward. The vertical velocity after time \( t \) can be calculated using the kinematic equation:
\[ v_y = u_y - g t \]
Substituting the expression for \( u_y \):
\[ v_y = u \sin \theta - g t \]

### Step 4: Calculate the change in momentum in the y-direction
The initial momentum in the y-direction is:
\[ p_{y, initial} = m u_y = m (u \sin \theta) \]
The final momentum in the y-direction is:
\[ p_{y, final} = m v_y = m (u \sin \theta - g t) \]
The change in momentum in the y-direction is:
\[ \Delta p_y = p_{y, final} - p_{y, initial} \]
Substituting the values:
\[ \Delta p_y = m (u \sin \theta - g t) - m (u \sin \theta) = -m g t \]

### Step 5: Determine the total change in momentum
Since the change in momentum in the x-direction is zero, the total change in momentum is simply the change in the y-direction:
\[ \Delta p = \Delta p_y = -m g t \]

### Step 6: Express the magnitude of the change in momentum
The magnitude of the change in momentum is:
\[ |\Delta p| = m g t \]

### Final Answer
The change in momentum of the body after time \( t \) is:
\[ |\Delta p| = m g t \]