| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Finding constants from motion conditions |
| Difficulty | Standard +0.3 This is a straightforward M2 kinematics question requiring standard techniques: forming simultaneous equations from given conditions, differentiating to find acceleration, integrating to find displacement, and calculating average speed. All steps are routine applications of calculus to motion formulas with no novel problem-solving required, making it slightly easier than average. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(500 = 25p + 5q\), \(12000 = 900p + 30q\) | B1 B1 | |
| Solve: \(750p = 9000\) | ||
| \(p = 12\), \(q = 40\) | M1 A1 (both) | |
| (b) Graph: parabola, increasing from \(t = 0\) | B2 | |
| (c) \(a = 24t + 40\) | M1 A1 | |
| \(t = 0\): \(a = 40\) m/s\(^{-2}\) (or their \(q\)) | ||
| (d) \(s = \int_0^{30} v \, dt = [4t^2 + 20t^2]_0^{30} = 126000\) m | M1 A1 M1 A1 | |
| (e) Travels a further \(20 \times 12000 = 240000\) m | B1 | |
| Average speed \(= 366000 \div 50 = 7320\) m/s\(^{-1}\) | M1 A1 | Total: 15 marks |
**(a)** $500 = 25p + 5q$, $12000 = 900p + 30q$ | B1 B1 |
Solve: $750p = 9000$ | |
$p = 12$, $q = 40$ | M1 A1 (both) |
**(b)** Graph: parabola, increasing from $t = 0$ | B2 |
**(c)** $a = 24t + 40$ | M1 A1 |
$t = 0$: $a = 40$ m/s$^{-2}$ (or their $q$) | |
**(d)** $s = \int_0^{30} v \, dt = [4t^2 + 20t^2]_0^{30} = 126000$ m | M1 A1 M1 A1 |
**(e)** Travels a further $20 \times 12000 = 240000$ m | B1 |
Average speed $= 366000 \div 50 = 7320$ m/s$^{-1}$ | M1 A1 | **Total: 15 marks**A rocket is fired from a fixed point $O$. During the first phase of its motion its velocity, $v$ ms$^{-1}$, is given at time $t$ seconds after firing by the formula
$$v = pt^2 + qt.$$
$5$ seconds after firing, the rocket is travelling at $500$ ms$^{-1}$.
$30$ seconds after firing, the rocket is travelling at $12\,000$ ms$^{-1}$.
\begin{enumerate}[label=(\alph*)]
\item Find the constants $p$ and $q$. [4 marks]
\item Sketch a velocity-time graph for the rocket for $0 \leq t \leq 30$. [2 marks]
\item Find the initial acceleration of the rocket. [2 marks]
\item Find the distance of the rocket from $O$ $30$ seconds after firing. [4 marks]
\end{enumerate}
From time $t = 30$ onwards, the rocket maintains a constant speed of $12\,000$ ms$^{-1}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{4}
\item Find the average speed of the rocket during its first $50$ seconds of motion. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 Q7 [15]}}