Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Convert the gravitational energy loss to kinetic-energy gain and use the positive root.
Worked solution
(a)
Convert the gravitational energy loss to kinetic-energy gain and use the positive root.
Conservation gives \(\displaystyle v^{2}=u^{2}+20h=324\), so the speed is \(\displaystyle \frac{18m}{s}\).
Answer: \(18\) m/s
(b)
Use conservation of mechanical energy between the two specified heights.
The lost gravitational potential energy is Mgh=\(\displaystyle \frac{897}{2}\) J.
Answer: \(\displaystyle \frac{897}{2}\) J
(c)
Set final speed to zero while retaining the initial kinetic energy at A.
Its initial kinetic energy supports extra rise \(\displaystyle \frac{u^{2}}{2g}=\)\(\displaystyle \frac{5}{4}\) m above A.
Answer: \(\displaystyle \frac{5}{4}\) m
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: Convert the gravitational energy loss to kinetic-energy gain and use the positive root.
- 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 vertical height drop, not an unspecified curved path length.
(b) · 2 marks
- 1 mark: Use conservation of mechanical energy between the two specified heights.
- 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: Initial kinetic energy allows the particle to rise above its original height.
(c) · 2 marks
- 1 mark: Set final speed to zero while retaining the initial kinetic energy at A.
- 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 vertical height drop, not an unspecified curved path length.
Total: 6 marks.