Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Use the first-stage force balance and its own elapsed time.
Worked solution
(a)
Use the first-stage force balance and its own elapsed time.
The first-stage resultant gives \(\displaystyle a= \frac{3m}{s^{2}}\). From rest over \(\displaystyle 3s, v= \frac{9m}{s}\).
Answer: \(9\) m/s
(b)
Use the driving-stage acceleration, not the later braking deceleration.
From rest with constant first-stage acceleration, s=at²/2=\(\displaystyle \frac{27}{2}\) m.
Answer: \(\displaystyle \frac{27}{2}\) m
(c)
The brake and continuing resistance remove kinetic energy. Their total opposing force is \(36\) N. Thus \(\displaystyle 36d=\frac{729}{2}\) gives \(\displaystyle d=\)\(\displaystyle \frac{81}{8}\) m.
Answer: \(\displaystyle \frac{81}{8}\) 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: Use the first-stage force balance and its own elapsed time.
- 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: Reset the stage time but keep the event velocity continuous.
(b) · 2 marks
- 1 mark: Use the driving-stage acceleration, not the later braking deceleration.
- 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: Do not carry the driving-stage acceleration into the braking stage.
(c) · 3 marks
- 1 mark: Identify all opposing work in the braking stage after the drive is removed.
- 1 mark: Equate that energy loss to the event kinetic energy and solve for a positive distance.
- 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: Reset the stage time but keep the event velocity continuous.
Total: 7 marks.