https://file.aiquickdraw.com/mathbot/file/1723214079036b1haoo4r.jpg

The image contains the following problem:

*A rifleman, who together with his rifle has a mass of 100 kg, stands on a smooth surface. He shoots 10 bullets horizontally. Each bullet has a mass of 10 g and a muzzle velocity of 800 m/s. What velocity does the man acquire after the last shot?*

To solve this problem, we can use the principle of conservation of momentum. Here's the step-by-step solution:

### Step 1: Understand the problem
- The total mass of the rifleman and the rifle: \( M_{\text{man}} = 100 \, \text{kg} \)
- Mass of each bullet: \( m_{\text{bullet}} = 10 \, \text{g} = 0.01 \, \text{kg} \)
- Muzzle velocity of each bullet: \( v_{\text{bullet}} = 800 \, \text{m/s} \)
- Number of bullets fired: 10

### Step 2: Apply the conservation of momentum
Since the surface is smooth (frictionless), the horizontal momentum is conserved. Initially, the total momentum is zero because both the man and the bullets are at rest.

After firing the bullets, the momentum of the bullets and the momentum of the man must sum to zero.

Let \( v_{\text{man}} \) be the velocity of the man after firing the bullets.

The momentum of one bullet after being fired is:
\[
p_{\text{bullet}} = m_{\text{bullet}} \times v_{\text{bullet}} = 0.01 \times 800 = 8 \, \text{kg}\cdot\text{m/s}
\]

The total momentum of all 10 bullets is:
\[
P_{\text{bullets}} = 10 \times p_{\text{bullet}} = 10 \times 8 = 80 \, \text{kg}\cdot\text{m/s}
\]

Since the total momentum is conserved, the momentum of the man (and rifle) must be equal in magnitude but opposite in direction to the total momentum of the bullets:
\[
M_{\text{man}} \times v_{\text{man}} = P_{\text{bullets}}
\]
\[
100 \times v_{\text{man}} = 80
\]

### Step 3: Solve for the velocity of the man
\[
v_{\text{man}} = \frac{80}{100} = 0.8 \, \text{m/s}
\]

### Final Answer:
The velocity of the man after the last shot is \( 0.8 \, \text{m/s} \) in the direction opposite to the bullet's motion.

Would you like more details or have any questions?

### Follow-up Questions:
1. How would the solution change if the surface had friction?
2. What would happen if the rifleman shot bullets in different directions?
3. How would the result differ if the bullets were fired with varying muzzle velocities?
4. How does the mass of the rifle affect the final velocity of the man?
5. What if the bullets were fired vertically rather than horizontally?
6. How would the problem change if only one bullet was fired?
7. What happens to the kinetic energy of the system after all the bullets are fired?
8. How does the number of bullets affect the final velocity?

### Tip:
Remember that conservation laws, like momentum conservation, are powerful tools for solving problems where no external forces are acting on the system.

Learn how to solve the problem of finding the velocity of a man after firing bullets using the principle of conservation of momentum. This physics problem involves calculating momentum, considering mass and velocity of bullets, and determining the final velocity of the man. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation