| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2018 |
| Session | March |
| Marks | 11 |
| Topic | Travel graphs |
| Type | Multi-stage motion with algebraic unknowns |
| Difficulty | Standard +0.3 This is a standard three-stage kinematics problem requiring a velocity-time graph sketch and application of SUVAT equations. While it involves algebraic manipulation to form and solve a quadratic, the approach is routine for M1 students: find times and distances for each stage, set up simultaneous equations from given constraints, and eliminate variables. The 'show that' structure guides students to the answer, and rejecting one root of the quadratic is straightforward. Slightly above average due to the algebraic manipulation required, but still a textbook-style question. |
| Spec | 3.02c Interpret kinematic graphs: gradient and area3.02d Constant acceleration: SUVAT formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Correct trapezium shape | B1 | Must start at the origin and stop on the \(t\) axis |
| Axes labelled correctly with \(V\) and 250 marked | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Acceleration 0.4 and deceleration 0.08 give time intervals of 2.5V and 12.5V respectively | B1 | Either of these intervals obtained |
| Three time intervals: 2.5V, 250 − 15V and 12.5V | M1 | Three time intervals found, in terms of \(V\), summing to 250; At least one interval must be correct |
| Area under their v-t graph = 880 | M1 | Attempt at area of trapezium or two triangles plus rectangle |
| \(\frac{1}{2} \times 2.5V \times V + V(250 - 15V) + \frac{1}{2} \times 12.5V \times V = 880\) | A1 | oe; unsimplified correct equation in \(V\) |
| Attempt simplification to required form | M1 | Dependent on both previous M marks; must correctly remove fractions in their equation and obtain result of the form \(aV^2 + bV + c = 0\) |
| \(3V^2 - 100V + 352 = 0\) | A1 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| \((3V - 88)(V - 4) = 0\) | M1 | Attempt to solve their 3-term quadratic; Or BC |
| \(V = \frac{88}{3}\) or \(V = 4\) | A1 | |
| \(V = 4\) only, as \(V = \frac{88}{3}\) is far too high (e.g. it equates to covering a distance of 100 m in less than 4 seconds) | A1 | Or 'mathematical' reason (e.g. time taken to decelerate from larger speed is greater than 250 s, so not possible) |
## Part (i):
Correct trapezium shape | B1 | Must start at the origin and stop on the $t$ axis
Axes labelled correctly with $V$ and 250 marked | B1 |
## Part (ii):
Acceleration 0.4 and deceleration 0.08 give time intervals of 2.5V and 12.5V respectively | B1 | Either of these intervals obtained
Three time intervals: 2.5V, 250 − 15V and 12.5V | M1 | Three time intervals found, in terms of $V$, summing to 250; At least one interval must be correct
Area under their v-t graph = 880 | M1 | Attempt at area of trapezium or two triangles plus rectangle
$\frac{1}{2} \times 2.5V \times V + V(250 - 15V) + \frac{1}{2} \times 12.5V \times V = 880$ | A1 | oe; unsimplified correct equation in $V$
Attempt simplification to required form | M1 | Dependent on both previous M marks; must correctly remove fractions in their equation and obtain result of the form $aV^2 + bV + c = 0$
$3V^2 - 100V + 352 = 0$ | A1 | AG
## Part (iii):
$(3V - 88)(V - 4) = 0$ | M1 | Attempt to solve their 3-term quadratic; Or BC
$V = \frac{88}{3}$ or $V = 4$ | A1 |
$V = 4$ only, as $V = \frac{88}{3}$ is far too high (e.g. it equates to covering a distance of 100 m in less than 4 seconds) | A1 | Or 'mathematical' reason (e.g. time taken to decelerate from larger speed is greater than 250 s, so not possible)A jogger is running along a straight horizontal road. The jogger starts from rest and accelerates at a constant rate of $0.4\,\text{m}\,\text{s}^{-2}$ until reaching a velocity of $V\,\text{m}\,\text{s}^{-1}$. The jogger then runs at a constant velocity of $V\,\text{m}\,\text{s}^{-1}$ before decelerating at a constant rate of $0.08\,\text{m}\,\text{s}^{-2}$ back to rest. The jogger runs a total distance of $880\,\text{m}$ in $250\,\text{s}$.
\begin{enumerate}[label=(\roman*)]
\item Sketch the velocity-time graph for the jogger's journey. [2]
\item Show that $3V^2 - 100V + 352 = 0$. [6]
\item Hence find the value of $V$, giving a reason for your answer. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/03 2018 Q8 [11]}}