Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Conserve signed momentum and use the combined final mass.
Worked solution
(a)
The momentum equation is \(3(7)+5(-6)=8V\). Thus \(\displaystyle V=\)\(\displaystyle - \frac{9}{8}\). Its sign gives the stated motion.
Answer: \(\displaystyle - \frac{9}{8}\) m/s on the right-positive axis; left.
(b)
Add signed momenta for the first quantity, but add the two positive half-mass-times-speed-squared terms for energy.
Answer: \(-9\) \(\displaystyle \frac{\mathrm{kg}m}{s}\) on the right-positive axis; \(\displaystyle \frac{327}{2}\) J.
(c)
Calculate the two initial energies and the single final energy separately.
Initial energy \(\displaystyle \frac{327}{2}\) minus the joined-particle energy \(\displaystyle \frac{81}{16}\) gives \(\displaystyle \frac{2535}{16}\) J.
Answer: \(\displaystyle \frac{2535}{16}\) J
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: Assign opposite signs to the initial velocities on one common axis.
- 1 mark: Use the combined mass and interpret the resulting final velocity sign.
- 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 the sum of the masses after the particles coalesce.
(b) · 3 marks
- 1 mark: Sum momenta with the opposing velocity signs.
- 1 mark: Calculate both initial kinetic energies and add them without cancelling opposing directions.
- 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: A negative final velocity is a direction, not an invalid speed.
(c) · 2 marks
- 1 mark: Calculate the two initial energies and the single final energy separately.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Use the sum of the masses after the particles coalesce.
Total: 8 marks.