Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Solve the velocity equation, order its roots and test signs.
Worked solution
(a)
The factored velocity has distinct roots 4 and 8. The positive leading coefficient makes it positive outside the two roots and negative between them.
Answer: 4 s and 8 s; signs +, −, + in chronological order.
(b)
An antiderivative of velocity is \(t^{3} - 18 t^{2} + 96 t\). Evaluation from 0 to 12 gives \(288\) m.
Answer: \(288\) m.
(c)
The signed integrals over the three intervals are \(160\), \(-32\), \(160\) m. The middle one is negative, so distance adds its magnitude to the other two, giving \(352\) m.
Answer: \(352\) 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) · 3 marks
- 1 mark: Solve \(\displaystyle v=0\) and retain both roots in the stated domain.
- 1 mark: Test or deduce the three interval signs and establish that both zeros reverse the motion.
- 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: Include both admissible velocity zeros, not only the first.
(b) · 2 marks
- 1 mark: Find a correct velocity antiderivative and evaluate it at both time endpoints.
- 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: Take the absolute value of each sign-consistent displacement before summing distance.
(c) · 3 marks
- 1 mark: Evaluate a separate signed displacement for each interval split by the two velocity zeros.
- 1 mark: Convert each interval contribution to a nonnegative distance and sum all three.
- 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: Include both admissible velocity zeros, not only the first.
Total: 8 marks.