A bullet is fired from a gun. The force on bullet is given by F = 600 – 2 × 10$^5$ t, where F is in Newton and t is in seconds. The force on the bullet is reduced to zero as soon as it leaves the barrel. The average impulse imparted to bullet is :

Question

Accepted Answer

To determine the average impulse imparted to the bullet, we need to calculate the impulse, which is the integral of force over time. The force on the bullet is given by the equation $ F(t) = 600 - 2 \times 10^5 \times t $.

1. **Calculate the time when the force becomes zero:**

The force becomes zero when:
   $$
   600 - 2 \times 10^5 \times t = 0
   $$
   Solving for $ t $:
   $$
   2 \times 10^5 \times t = 600
   $$
   $$
   t = \frac{600}{2 \times 10^5} = 3 \times 10^{-3} \, \text{seconds}
   $$

2. **Calculate the impulse:**

Impulse $ J $ is given by the integral of force over time:
   $$
   J = \int_0^{3 \times 10^{-3}} (600 - 2 \times 10^5 \times t) \, dt
   $$

Evaluating the integral:
   $$
   J = \left[ 600t - \frac{2 \times 10^5 \times t^2}{2} \right]_0^{3 \times 10^{-3}}
   $$
   $$
   J = \left[ 600 \times 3 \times 10^{-3} - 10^5 \times (3 \times 10^{-3})^2 \right]
   $$
   $$
   J = \left[ 1.8 - 10^5 \times 9 \times 10^{-6} \right]
   $$
   $$
   J = \left[ 1.8 - 0.9 \right]
   $$
   $$
   J = 0.9 \, \text{Ns}
   $$

3. **Explanation of why Option 2 is correct:**

The calculated impulse is 0.9 Ns, which matches Option 2. This confirms that the average impulse imparted to the bullet is indeed 0.9 Ns.

4. **Explanation of why other options are incorrect:**

- **Option 1 (Zero):** The impulse cannot be zero because there is a non-zero force acting on the bullet over a finite time interval, resulting in a non-zero impulse.
   - **Option 3 (1.8 Ns):** This value is incorrect because it does not match the calculated impulse of 0.9 Ns. It seems to be a miscalculation or misunderstanding of the integral.
   - **Option 4 (9.0 Ns):** This value is significantly higher than the calculated impulse and does not correspond to the result of the integral of the given force function.

In conclusion, the correct answer is Option 2, as the calculated impulse imparted to the bullet is 0.9 Ns.