More practice questions

4. Q4. A gun initially at rest and free to recoil has mass 4 kg and fires a bullet of mass 20 g; neglect propellant-gas momentum. The bullet leaves the barrel at 402 m s^⁻¹ relative to the gun. Find the speed of the bullet relative to the ground and the recoil speed of the gun.

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   Let the bullet move at v (forward) and the gun at V (backward), both relative to the ground. Momentum: 0.02 v = 4 V. Relative speed: v + V = 402. So V = 0.005 v and 1.005 v = 402, giving v = 400 m s⁻¹ and V = 2 m s⁻¹.

5. Q5. Two skaters of masses 50 kg and 75 kg stand at rest on smooth ice and push each other apart. The 50 kg skater moves off at 1.5 m s^⁻¹. Find the velocity of the other skater.

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   $$\begin{aligned} 50 \times 1.5 + 75\,v &= 0 \\ v &= -\frac{75}{75} \\ &= -1\ \text{m s}^{-1} \end{aligned}$$ The 75 kg skater moves at 1 m s⁻¹ in the opposite direction.

6. Q6. A machine gun fires 10 bullets per second, each of mass 40 g, at 500 m s^⁻¹. What average force must the gunner apply to hold the gun steady?

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   Each second the gun gives the bullets a momentum of

   $$10 \times 0.04 \times 500 = 200\ \text{kg m s}^{-1}$$ By the second law, the average force of the gun on the bullets is this rate of change of momentum, 200 N. By the third law, the bullets push the gun back with 200 N, so the gunner must push forward with 200 N to hold it steady.

7. Q7. A ball falls freely towards the ground. Is its momentum conserved? Is there any system whose momentum is conserved here?

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   The ball’s momentum is not conserved: the Earth’s gravity is an external force on it and its momentum increases downward. For the system of the ball and the Earth, the gravitational forces are internal and equal and opposite, so the total momentum of ball plus Earth is conserved. Neglecting other external interactions, the Earth gains an equal upward change in momentum, but its speed change is far too small to notice.