June 2023 Paper 2 Q14
14 A car has an initial velocity of 1 m s−1
The car is moving in a straight line.
The acceleration \(a\) m s−2 of the car at time \(t\) seconds is given by
\[a = 3kt^2 - 2kt + 1\]where \(k\) is a constant.
When \(t = 3\) the car has a velocity of 10 m s−1
Show that \(k = \dfrac{1}{3}\) [4 marks]
| Scheme | Marks | AO |
|---|---|---|
| Integrates \(a\) with at least one term correct. | M1 | 3.4 |
| Obtains a fully correct expression for \(v\) ACF Coefficients can be unsimplified. Condone omission of constant | A1 | 1.1b |
| Uses given initial conditions to find their constant of integration. This must be done before \(v\) = 10 and \(t\) = 3 are substituted. | M1 | 3.4 |
| Completes reasoned argument by substituting \(v\) = 10 and \(t\) = 3 into \(v = kt^3 - kt^2 + t + 1\) to show \(k = \dfrac{1}{3}\) Must include at least more one intermediate step after substituting. AG | A1 | 1.1b |
| (4 marks) |
Typical solution
\[v = \int a\,\mathrm{d}t\]\[v = kt^3 - kt^2 + t + c\]\(v = 1\) when \(t = 0\) then \(c = 1\)
\(v\) = 10 and \(t\) = 3
\[10 = 27k - 9k + 3 + 1\]\[18k = 6\]\[k = \frac{1}{3}\]