A disc of mass 10 g is kept floating horizontal in the air by firing bullets, each of mass 5g, with the same velocity at the same rate of 10 bullets per second. The bullets rebound with the same speed in positive direction . The velocity of each bullet at the time of impact is (Take `g = 9.8 ms^(-2))`

Options

A. `196 cm s^(-1)`
B. `98 cm s^(-1)`
C. `49 cm s^(-1)`
D. `392 cm s^(-1)`

Answer

To solve the problem step by step, we will analyze the forces acting on the disc and the change in momentum of the bullets.

### Step 1: Identify the forces acting on the disc
The disc has a weight acting downwards due to gravity, which can be calculated as:
\[ \text{Weight} = M \cdot g \]
where \( M = 10 \, \text{g} = 0.01 \, \text{kg} \) (converting grams to kilograms) and \( g = 9.8 \, \text{m/s}^2 \).

### Step 2: Calculate the weight of the disc
Substituting the values:
\[ \text{Weight} = 0.01 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 0.098 \, \text{N} \]

### Step 3: Determine the upward force provided by the bullets
Each bullet has a mass \( m = 5 \, \text{g} = 0.005 \, \text{kg} \) and they are fired at a rate of 10 bullets per second. The change in momentum for each bullet when it hits the disc and rebounds can be calculated as:
\[ \Delta p = m \cdot (v_f - v_i) \]
where \( v_f \) is the final velocity (upward) and \( v_i \) is the initial velocity (downward). Since the bullets rebound with the same speed, we can express this as:
\[ \Delta p = m \cdot (v - (-v)) = m \cdot (v + v) = 2mv \]

### Step 4: Calculate the total change in momentum for 10 bullets
Since 10 bullets are fired per second, the total change in momentum per second is:
\[ \Delta P = 10 \cdot (2mv) = 20mv \]

### Step 5: Set the upward force equal to the weight of the disc
For the disc to remain floating, the upward force must equal the weight of the disc:
\[ 20mv = Mg \]
Substituting the values:
\[ 20 \cdot 0.005 \cdot v = 0.01 \cdot 9.8 \]

### Step 6: Solve for \( v \)
Now, we can simplify and solve for \( v \):
\[ 0.1v = 0.098 \]
\[ v = \frac{0.098}{0.1} = 0.98 \, \text{m/s} \]

### Step 7: Convert the velocity to cm/s
Since the question asks for the velocity in cm/s:
\[ v = 0.98 \, \text{m/s} \times 100 = 98 \, \text{cm/s} \]

### Final Answer
The velocity of each bullet at the time of impact is:
\[ \boxed{98 \, \text{cm/s}} \]