Hint
Choose and retain a positive direction; distinguish signed quantities from magnitudes. Integrate acceleration and determine the constant from the initial velocity.
Worked solution
(a)
An antiderivative of acceleration is \(3 t^{2} + 4 t\)+C. The initial condition gives \(\displaystyle C=7\).
Answer: \(v=3 t^{2} + 4 t + 7\) m/s, with t in seconds.
(b)
Subtract the initial velocity from the evaluated velocity or integrate acceleration definitely.
The definite integral of acceleration gives \(64\) m/s.
Answer: \(64\) m/s
(c)
Integrate the recovered velocity, including its initial-velocity term.
Integrating the complete velocity rule from 0 to 4 gives \(124\) m.
Answer: \(124\) 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: Integrate the entire acceleration expression.
- 1 mark: Substitute the initial velocity into the antiderivative to determine its constant.
- 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: The initial velocity is not a position constant.
(b) · 2 marks
- 1 mark: Subtract the initial velocity from the evaluated velocity or integrate acceleration definitely.
- 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 second integration needs its own initial position if absolute position is requested.
(c) · 2 marks
- 1 mark: Integrate the recovered velocity, including its initial-velocity term.
- 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: The initial velocity is not a position constant.
Total: 7 marks.