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, we need to analyze the forces acting on the disc and the effect of the bullets being fired at it. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Forces Acting on the Disc
The disc has a mass of 10 g, which means it experiences a downward gravitational force (weight) given by:
\[ F_g = m \cdot g \]
where \( m = 10 \, \text{g} = 0.01 \, \text{kg} \) and \( g = 9.8 \, \text{m/s}^2 \).

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

### Step 3: Determine the Momentum Change Due to Bullets
Each bullet has a mass of 5 g (0.005 kg) and they are fired at a rate of 10 bullets per second. When a bullet hits the disc and rebounds with the same speed, the change in momentum for one bullet is:
\[ \Delta p = m \cdot (v - (-v)) = m \cdot (v + v) = 2mv \]
where \( v \) is the velocity of the bullet just before impact.

### Step 4: Calculate the Total Force Exerted by the Bullets
The total change in momentum per second (which is the force exerted by the bullets) is:
\[ F_b = \text{Rate of bullets} \cdot \Delta p = 10 \cdot (2 \cdot 0.005 \cdot v) = 0.1v \, \text{N} \]

### Step 5: Set the Forces Equal to Each Other
For the disc to float, the upward force due to the bullets must equal the downward gravitational force:
\[ F_b = F_g \]
Thus:
\[ 0.1v = 0.098 \]

### Step 6: Solve for the Velocity \( v \)
Rearranging the equation to find \( v \):
\[ v = \frac{0.098}{0.1} = 0.98 \, \text{m/s} \]

### Step 7: Convert the Velocity to cm/s
To express the velocity in centimeters per second:
\[ v = 0.98 \, \text{m/s} \times 100 = 98 \, \text{cm/s} \]

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