a 100kg gun fires a ball of 1kg horizontally from a cliff of height 500m. If falls on the ground at a distance of 400m from the bottom of the cliff. The recoil velocity of the gun is (Take g: `10ms^(-2)`

Options

A. `0.2"ms"^(-1)`
B. `0.4"ms"^(-1)`
C. `0.6"ms"^(-1)`
D. `0.8"ms"^(-1)`

Explanation

To solve the problem of finding the recoil velocity of the gun when it fires a ball horizontally from a cliff, we can follow these steps:

### Step 1: Identify the given data
- Mass of the gun, \( M_g = 100 \, \text{kg} \)
- Mass of the ball, \( M_b = 1 \, \text{kg} \)
- Height of the cliff, \( h = 500 \, \text{m} \)
- Horizontal distance from the base of the cliff where the ball lands, \( d = 400 \, \text{m} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)

### Step 2: Calculate the time of flight of the ball
The time \( t \) it takes for the ball to fall from the height of the cliff can be calculated using the formula for free fall:
\[ h = \frac{1}{2} g t^2 \]
Rearranging this gives:
\[ t = \sqrt{\frac{2h}{g}} \]
Substituting the values:
\[ t = \sqrt{\frac{2 \times 500}{10}} = \sqrt{100} = 10 \, \text{s} \]

### Step 3: Calculate the horizontal velocity of the ball
The horizontal distance \( d \) traveled by the ball can be expressed as:
\[ d = u \cdot t \]
Where \( u \) is the horizontal velocity of the ball. Rearranging gives:
\[ u = \frac{d}{t} \]
Substituting the values:
\[ u = \frac{400}{10} = 40 \, \text{m/s} \]

### Step 4: Apply conservation of momentum
Since there are no external horizontal forces acting on the system, the momentum before firing must equal the momentum after firing. Before firing, the total momentum is zero. After firing:
\[ M_b \cdot u + M_g \cdot v_g = 0 \]
Where \( v_g \) is the recoil velocity of the gun. Rearranging gives:
\[ M_g \cdot v_g = -M_b \cdot u \]
Thus,
\[ v_g = -\frac{M_b \cdot u}{M_g} \]

### Step 5: Substitute the known values to find \( v_g \)
Substituting the values:
\[ v_g = -\frac{1 \cdot 40}{100} = -0.4 \, \text{m/s} \]

### Conclusion
The recoil velocity of the gun is \( -0.4 \, \text{m/s} \). The negative sign indicates that the direction of the recoil is opposite to the direction of the ball's motion.