| Exam Board | OCR MEI |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Displacement from velocity by integration |
| Difficulty | Moderate -0.8 This is a straightforward mechanics question requiring basic differentiation of a quadratic to find acceleration, and integration to find distance. Part (i) is graph interpretation (curved line means non-constant acceleration), part (ii) is routine differentiation and interpretation, and part (iii) is standard definite integration. All techniques are elementary calculus with no problem-solving insight required. |
| Spec | 1.07b Gradient as rate of change: dy/dx notation1.08e Area between curve and x-axis: using definite integrals3.02b Kinematic graphs: displacement-time and velocity-time3.02c Interpret kinematic graphs: gradient and area |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The line is not straight | B1 | Any valid comment |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a = 3 - \frac{6t}{8}\) | M1 | Attempt to differentiate. Accept 1 term correct but not \(3 - \frac{3t}{8}\) |
| \(a(4) = 0\) | F1 | |
| The sprinter has reached a steady speed | E1 | Accept 'stopped accelerating' but not just \(a = 0\). Do not FT \(a(4) \neq 0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int_1^4 \left(3t - \frac{3t^2}{8}\right)dt\) | M1 | Integrating. Neglect limits |
| \(= \left[\frac{3t^2}{2} - \frac{t^3}{8}\right]_1^4\) | A1 | One term correct. Neglect limits |
| \(= (24 - 8) - \left(\frac{3}{2} - \frac{1}{8}\right)\) | M1 | Correct limits substituted in integral. Subtraction seen. If arb constant used, evaluated to give \(s=0\) when \(t=1\) and then sub \(t=4\) |
| \(= 14\frac{5}{8}\) m \((14.625\) m\()\) | A1 | cao. Any form. [If trapezium rule used: M1 use of rule (must be clear method and at least two regions), A1 correctly applied, M1 at least 6 regions used, A1 answer correct to at least 2 s.f.] |
## Question 5:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| The line is not straight | B1 | Any valid comment |
**Subtotal: 1 mark**
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a = 3 - \frac{6t}{8}$ | M1 | Attempt to differentiate. Accept 1 term correct but not $3 - \frac{3t}{8}$ |
| $a(4) = 0$ | F1 | |
| The sprinter has reached a steady speed | E1 | Accept 'stopped accelerating' but not just $a = 0$. Do not FT $a(4) \neq 0$ |
**Subtotal: 3 marks**
### Part (iii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_1^4 \left(3t - \frac{3t^2}{8}\right)dt$ | M1 | Integrating. Neglect limits |
| $= \left[\frac{3t^2}{2} - \frac{t^3}{8}\right]_1^4$ | A1 | One term correct. Neglect limits |
| $= (24 - 8) - \left(\frac{3}{2} - \frac{1}{8}\right)$ | M1 | Correct limits substituted in integral. Subtraction seen. If arb constant used, evaluated to give $s=0$ when $t=1$ and then sub $t=4$ |
| $= 14\frac{5}{8}$ m $(14.625$ m$)$ | A1 | cao. Any form. [If trapezium rule used: M1 use of rule (must be clear method and at least two regions), A1 correctly applied, M1 at least 6 regions used, A1 answer correct to at least 2 s.f.] |
**Subtotal: 4 marks**
**Total: 8 marks**
---5 Fig. 3 is a sketch of the velocity-time graph modelling the velocity of a sprinter at the start of a race.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{34e4ce80-21b0-48f5-865c-de4dd837f7c5-4_581_1085_453_567}
\captionsetup{labelformat=empty}
\caption{Fig. 3}
\end{center}
\end{figure}
(i) How can you tell from the sketch that the acceleration is not modelled as being constant for $0 \leqslant t \leqslant 4$ ?
The velocity of the sprinter, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, for the time interval $0 \leqslant t \leqslant 4$ is modelled by the expression
$$v = 3 t - \frac { 3 } { 8 } t ^ { 2 } .$$
(ii) Find the acceleration that the model predicts for $t = 4$ and comment on what this suggests about the running of the sprinter.\\
(iii) Calculate the distance run by the sprinter from $t = 1$ to $t = 4$.
\hfill \mbox{\textit{OCR MEI M1 Q5 [8]}}