Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Use gravitational force times the vertical drop.
Worked solution
(a)
Use gravitational force times the vertical drop.
The vertical drop is L sin α, giving positive work \(\displaystyle \frac{1200}{13}\) J.
Answer: \(\displaystyle \frac{1200}{13}\) J
(b)
Use the actual along-plane travel distance for frictional energy loss.
The opposing force μMg cos α acts over distance L, removing \(\displaystyle \frac{576}{13}\) J.
Answer: \(\displaystyle \frac{576}{13}\) J
(c)
Gravity supplies \(\displaystyle \frac{1200}{13}\) J and friction removes \(\displaystyle \frac{576}{13}\) J. Thus \(2v^2=48\) gives the positive speed \(4.90\) m/s.
Answer: \(4.90\) m/s.
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) · 2 marks
- 1 mark: Use gravitational force times the vertical drop.
- 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: Friction removes mechanical energy even when gravity increases kinetic energy overall.
(b) · 2 marks
- 1 mark: Use the actual along-plane travel distance for frictional energy loss.
- 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: A stated height drop and an along-slope distance use different trigonometric factors.
(c) · 3 marks
- 1 mark: Subtract the frictional loss from gravitational work.
- 1 mark: Use the kinetic-energy change from rest and select a positive speed with the stated rounding.
- 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: Friction removes mechanical energy even when gravity increases kinetic energy overall.
Total: 7 marks.