| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Travel graphs |
| Type | Multi-stage motion with all parameters given |
| Difficulty | Standard +0.8 This question combines standard kinematics (parts a-b) with calculus-based mechanics requiring integration and constraint solving (parts c-d). While the initial parts are routine M1 content, parts (c)-(d) require students to work with a variable acceleration model, find constants from given conditions (maximum at t=5, specific values), and apply calculus—significantly elevating difficulty above typical M1 questions which focus on constant acceleration. |
| Spec | 1.08l Interpret differential equation solutions: in context3.02d Constant acceleration: SUVAT formulae |
| Answer | Marks | Guidance |
|---|---|---|
| (a) [Velocity-time graph: trapezoid with vertices at \((0, 0)\), \((10, 40)\), \((60, 40)\), \((72, 0)\)] | B3 | |
| (b) Time for last section \(= 240 + \frac{1}{2}(40) = 12 \text{ s}\), so total time \(= 72 \text{ s}\) | M1 A1 | |
| Total distance \(= \frac{1}{2}(50 + 72) \times 40 = 2440 \text{ m}\) | M1 A1 | |
| Average speed \(= 2440 \div 72 = 33.9 \text{ m s}^{-1}\) | M1 A1 | |
| (c) Put \(t = 5\): \(k(5m - 25) = 4\) | Put \(t = 10\): \(k(10m - 100) = 0\) | B1 B1 |
| \(k = \frac{4}{25}, m = 10\) | M1 A1 A1 | |
| (d) When \(t = 2\), \(a = \frac{4}{25} \times 16 = 2.56 \text{ m s}^{-2}\) | M1 A1 | 16 marks |
(a) [Velocity-time graph: trapezoid with vertices at $(0, 0)$, $(10, 40)$, $(60, 40)$, $(72, 0)$] | B3
(b) Time for last section $= 240 + \frac{1}{2}(40) = 12 \text{ s}$, so total time $= 72 \text{ s}$ | M1 A1
Total distance $= \frac{1}{2}(50 + 72) \times 40 = 2440 \text{ m}$ | M1 A1
Average speed $= 2440 \div 72 = 33.9 \text{ m s}^{-1}$ | M1 A1
(c) Put $t = 5$: $k(5m - 25) = 4$ | Put $t = 10$: $k(10m - 100) = 0$ | B1 B1
$k = \frac{4}{25}, m = 10$ | M1 A1 A1
(d) When $t = 2$, $a = \frac{4}{25} \times 16 = 2.56 \text{ m s}^{-2}$ | M1 A1 | **16 marks**A car starts from rest at time $t = 0$ and moves along a straight road with constant acceleration 4 ms$^{-2}$ for 10 seconds. It then travels at a constant speed for 50 seconds before decelerating to rest over a further distance of 240 m.
\begin{enumerate}[label=(\alph*)]
\item Sketch a graph of velocity against time for the total period of the car's motion. [3 marks]
\item Find the car's average speed for the whole journey. [6 marks]
\end{enumerate}
In reality the car's acceleration $a$ ms$^{-2}$ in the first 10 seconds is not constant, but increases from 0 to 4 ms$^{-2}$ in the first 5 seconds and then decreases to 0 again. A refined model designed to take account of this uses the formula $a = k(mt - t^2)$ for $0 \leq t \leq 10$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Calculate the values of the constants $k$ and $m$. [5 marks]
\item Find the acceleration of the car when $t = 2$ according to this model. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 Q7 [16]}}