More practice questions

4. Q4. Two 1 kg balls move along a line at 3 m s^⁻¹ and 1 m s^⁻¹ in the same direction. Find the total kinetic energy, the kinetic energy of the centre of mass, and the kinetic energy about the centre of mass.

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   Total: K = ½(1)(3^²) + ½(1)(1^²) = 4.5 + 0.5 = 5 J. The centre of mass moves at V = (3 + 1)/2 = 2 m s^⁻¹, so ½MV^² = ½(2)(2^²) = 4 J. Relative to the centre of mass the balls move at +1 and −1 m s^⁻¹, so K′ = 0.5 + 0.5 = 1 J. Check: 1 + 4 = 5 J.

5. Q5. An astronaut of mass 60 kg floats at rest in space and throws a 2 kg tool at 6 m s^⁻¹ (relative to the frame in which both were at rest). How fast does the astronaut move afterwards, and what happens to the centre of mass of astronaut and tool?

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   The total momentum stays zero: 60v + 2(6) = 0, so v = −0.2 m s^⁻¹. The astronaut drifts back at 0.2 m s^⁻¹, and the centre of mass of the two stays at rest.

6. Q6. A ball falls freely towards the Earth. Is the momentum of the ball conserved? Is the momentum of the ball and the Earth together conserved?

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   The ball’s momentum is not conserved: the Earth’s gravity is an external force on the ball, and its momentum grows. For the ball and the Earth together, gravity is internal (the ball pulls the Earth up exactly as hard as the Earth pulls the ball down), so the total momentum is conserved when other external interactions are neglected. The Earth’s share of the motion is far too small to notice because its mass is so large.