Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Conserve total signed momentum across the collision.
Worked solution
(a)
Momentum gives B’s velocity \(\displaystyle \frac{48}{5}\) m/s. It is greater than A’s final rightward velocity \(\displaystyle \frac{24}{5}\) m/s, so B, which is ahead, moves away from A.
Answer: \(\displaystyle \frac{48}{5}\) m/s right; they separate.
(b)
Before impact B is stationary. After impact use each particle’s own mass and final speed in a separate kinetic-energy term, then add.
Answer: Before: \(1440\) J; after: \(\displaystyle \frac{2592}{5}\) J.
(c)
The loss is \(\displaystyle \frac{4608}{5}\) J out of \(1440\) J initially. The positive loss establishes that total kinetic energy is not conserved, even though total momentum is. The lost kinetic energy is transferred to other forms, such as internal energy; this does not contradict conservation of total energy.
Answer: \(64\)%; the collision loses kinetic energy.
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternative methods and consequential reasoning. An unexplained result does not earn a method mark.
(a) · 3 marks
- 1 mark: Use conservation of momentum with the given final velocity.
- 1 mark: Compare the two post-impact velocities in the stated spatial order.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not assume kinetic energy is conserved in a general collision.
(b) · 3 marks
- 1 mark: Calculate the initial energy from the one initially moving particle.
- 1 mark: Calculate and sum both final kinetic energies using the recovered velocity.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Use one fixed positive direction before and after impact.
(c) · 3 marks
- 1 mark: Calculate the energy difference and divide it by the initial energy.
- 1 mark: Interpret the nonzero loss without confusing momentum conservation with kinetic-energy conservation.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not assume kinetic energy is conserved in a general collision.
Total: 9 marks.