Explanation:

To find the recoil velocity of the rifle when a bullet is fired, we can use the law of conservation of momentum. The total momentum before the shot must equal the total momentum after the shot.

Step 1: Calculate the initial momentum of the system (bullet + rifle) before the shot.

Momentum (p) = mass (m) × velocity (v)

For the bullet:

Mass of the bullet (m_bullet) = 10 g = 0.01 kg

Initial velocity of the bullet (v_bullet) = 250 m/s

Momentum of the bullet (p_bullet) = m_bullet × v_bullet

For the rifle:

Mass of the rifle (m_rifle) = 5 kg

Initial velocity of the rifle (v_rifle) = 0 m/s (since it's at rest)

Momentum of the rifle (p_rifle) = m_rifle × v_rifle

The total initial momentum of the system is the sum of the bullet and the rifle's momenta:

Total initial momentum = p_bullet + p_rifle

Step 2: Calculate the final momentum of the system after the shot.

After the bullet is fired, both the bullet and the rifle will be in motion, and we need to find the final momentum of the system. The bullet's momentum is given by its mass (0.01 kg) and the velocity it acquires after being fired (let's call it v_bullet_f), and the rifle's momentum is given by its mass (5 kg) and the velocity it acquires in the opposite direction (let's call it v_rifle_f).

Total final momentum = p_bullet_f + p_rifle_f

Step 3: Apply the conservation of momentum.

According to the law of conservation of momentum, the total initial momentum is equal to the total final momentum:

Total initial momentum = Total final momentum

p_bullet + p_rifle = p_bullet_f + p_rifle_f

Step 4: Solve for v_rifle_f.

Since the bullet and rifle move in opposite directions after the shot, we need to consider their velocities as being in opposite directions. Therefore, the final momentum of the bullet (p_bullet_f) is in the negative direction, and the final momentum of the rifle (p_rifle_f) is in the positive direction.

p_rifle_f - p_bullet_f = p_bullet + p_rifle

Now, we can plug in the values:

m_rifle × v_rifle_f - (m_bullet × v_bullet_f) = (m_bullet × v_bullet) + (m_rifle × v_rifle)

Solve for v_rifle_f:

v_rifle_f = [(m_bullet × v_bullet) + (m_rifle × v_rifle)] / m_rifle

Step 5: Calculate v_rifle_f.

v_rifle_f = [(0.01 kg × 250 m/s) + (5 kg × v_rifle)] / 5 kg

v_rifle_f = (2.5 kg·m/s + 5 kg·v_rifle) / 5 kg

v_rifle_f = (2.5 + v_rifle) m/s

Now, you have an equation for v_rifle_f in terms of v_rifle. To find the recoil velocity of the rifle (v_rifle_f), you can solve for v_rifle_f:

v_rifle_f = 2.5 m/s + v_rifle

This is the expression for the recoil velocity of the rifle after the bullet is fired.

Verified answer

Explanation:

To find the recoil velocity of the rifle when a bullet is fired, you can use the law of conservation of momentum. The total momentum before firing must be equal to the total momentum after firing.

The momentum of the bullet before firing is:

Momentum_bullet_before = (mass_bullet) * (velocity_bullet) = (0.01 kg) * (0 m/s) = 0 kg m/s

The momentum of the rifle before firing is:

Momentum_rifle_before = (mass_rifle) * (velocity_rifle) = (5 kg) * (0 m/s) = 0 kg m/s

The total momentum before firing is 0 kg m/s.

When the bullet is fired, its momentum changes, and the rifle's momentum changes in the opposite direction to compensate for it. The total momentum after firing must also be 0 kg m/s.

Let V be the recoil velocity of the rifle after firing. The bullet's momentum after firing is:

Momentum_bullet_after = (mass_bullet) * (velocity_bullet) = (0.01 kg) * (250 m/s) = 2.5 kg m/s

Now, you set up the equation for the total momentum after firing:

0 kg m/s (rifle) + 2.5 kg m/s (bullet) = 0 kg m/s (total)

0 + 2.5 = 0

2.5 kg m/s = 0

Since the total momentum must be zero after firing, the recoil velocity of the rifle must also be 0 m/s. So, the recoil velocity of the rifle is 0 m/s.