Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Differentiate each time-dependent position term before evaluation.
Worked solution
(a)
Differentiate each time-dependent position term before evaluation.
The derivative \(3 t^{2} + 5\) gives \(17\) m/s at the stated time.
Answer: \(17\) m/s
(b)
Differentiate the position twice with respect to time.
Differentiating the velocity gives \(6 t\) and hence \(12\) \(\displaystyle \frac{m}{s^{2}}\).
Answer: \(12\) \(\displaystyle \frac{m}{s^{2}}\)
(c)
Calculate the difference of positions, including the origin constant consistently.
Subtract \(\displaystyle x(0)=7\) from the final position to obtain \(18\) m.
Answer: \(18\) 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: Differentiate each time-dependent position term before evaluation.
- 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 differentiate after first replacing t by a number.
(b) · 2 marks
- 1 mark: Differentiate the position twice with respect to 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: A constant initial position has zero derivative but remains in position evaluations.
(c) · 2 marks
- 1 mark: Calculate the difference of positions, including the origin constant consistently.
- 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 differentiate after first replacing t by a number.
Total: 6 marks.