A shell of mass 100 kg is fired horizontally from a cannon with a mass of `5xx10^(3) kg`. The kinetic energy of the shell at the end of the barrel is `7.5xx10^(6) J`. The kinetic energy imparted to the cannon by the recoil is

Options

A. `2xx10^(5) J`
B. `7.5 xx10^(6) J`
C. `1.5 xx10^(5) J`
D. `10^(5)J`

Explanation

To solve the problem, we need to find the kinetic energy imparted to the cannon by the recoil when a shell is fired from it. We will use the principle of conservation of momentum and the relationship between kinetic energy and momentum.

### Step-by-Step Solution:

1. **Identify the given values:**
- Mass of the shell, \( m = 100 \, \text{kg} \)
- Mass of the cannon, \( M = 5 \times 10^3 \, \text{kg} = 5000 \, \text{kg} \)
- Kinetic energy of the shell, \( KE_{\text{shell}} = 7.5 \times 10^6 \, \text{J} \)

2. **Use the conservation of momentum:**
- Initially, the system (cannon + shell) is at rest, so the total initial momentum is zero.
- When the shell is fired, it moves in one direction, and the cannon recoils in the opposite direction.
- Let \( P_{\text{shell}} \) be the momentum of the shell and \( P_{\text{cannon}} \) be the momentum of the cannon.
- According to conservation of momentum:
\[ P_{\text{shell}} + P_{\text{cannon}} = 0 \implies P_{\text{cannon}} = -P_{\text{shell}} \]

3. **Relate kinetic energy to momentum:**
- The kinetic energy of an object is given by:
\[ KE = \frac{P^2}{2m} \]
- For the shell:
\[ KE_{\text{shell}} = \frac{P_{\text{shell}}^2}{2m} \]
- For the cannon:
\[ KE_{\text{cannon}} = \frac{P_{\text{cannon}}^2}{2M} \]

4. **Express \( P_{\text{cannon}} \) in terms of \( P_{\text{shell}} \):**
- Since \( P_{\text{cannon}} = -P_{\text{shell}} \), we have:
\[ KE_{\text{cannon}} = \frac{P_{\text{shell}}^2}{2M} \]

5. **Relate the kinetic energies of the shell and cannon:**
- From the equations for kinetic energy, we can derive:
\[ \frac{KE_{\text{cannon}}}{KE_{\text{shell}}} = \frac{P_{\text{cannon}}^2 / 2M}{P_{\text{shell}}^2 / 2m} = \frac{m}{M} \]
- Substituting the values:
\[ \frac{KE_{\text{cannon}}}{7.5 \times 10^6} = \frac{100}{5000} = \frac{1}{50} \]

6. **Calculate the kinetic energy of the cannon:**
- Therefore:
\[ KE_{\text{cannon}} = \frac{7.5 \times 10^6}{50} = 1.5 \times 10^5 \, \text{J} \]

### Final Answer:
The kinetic energy imparted to the cannon by the recoil is \( 1.5 \times 10^5 \, \text{J} \).

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