| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Force depends on velocity v |
| Difficulty | Challenging +1.8 This M3 question requires setting up and solving a separable differential equation with a time-dependent resistance force, then analyzing the resulting expression for extrema. The separation of variables is non-standard due to the sin(t/100) term, requiring integration techniques beyond routine mechanics problems. While the mathematical steps are systematic, the combination of variable resistance, trigonometric integration, and optimization makes this significantly harder than typical M3 questions but not exceptionally difficult for the module. |
| Spec | 6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{dv}{dt} = -v^2 \sin\left(\frac{t}{100}\right)\) | M1 A1 | |
| \(\int \frac{1}{v} dv = -\int \sin\left(\frac{t}{100}\right) dt\) | A1 M1 A1 | |
| \(-\frac{1}{v} = 100 \cos\left(\frac{t}{100}\right) + c\) | ||
| \(t = 0, v = 0.2\); \(c = -105\) | ||
| \(\frac{1}{v} = 105 - 100 \cos\left(\frac{t}{100}\right)\) | M1 A1 | |
| (b) \(v_{\max} = 0.2 \text{ ms}^{-1}\) (initial speed) | M1 A1 A1 | |
| \(v_{\min} = 0.00952 \text{ ms}^{-1}\) (\(t = 50\pi\)) | Total: 10 marks |
**(a)** $\frac{dv}{dt} = -v^2 \sin\left(\frac{t}{100}\right)$ | M1 A1 |
$\int \frac{1}{v} dv = -\int \sin\left(\frac{t}{100}\right) dt$ | A1 M1 A1 |
$-\frac{1}{v} = 100 \cos\left(\frac{t}{100}\right) + c$ | |
$t = 0, v = 0.2$; $c = -105$ | |
$\frac{1}{v} = 105 - 100 \cos\left(\frac{t}{100}\right)$ | M1 A1 |
**(b)** $v_{\max} = 0.2 \text{ ms}^{-1}$ (initial speed) | M1 A1 A1 |
$v_{\min} = 0.00952 \text{ ms}^{-1}$ ($t = 50\pi$) | | **Total: 10 marks**Suraiya, whose mass is $m$ kg, takes a running jump into a swimming pool so that she begins to swim in a straight line with speed 0·2 ms$^{-1}$. She continues to move in the same straight line, the only force acting on her being a resistance of magnitude $mv^2 \sin \left(\frac{t}{100}\right)$ N, where $v$ ms$^{-1}$ is her speed at time $t$ seconds after entering the pool and $0 \leq t \leq 50\pi$.
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $v$ in terms of $t$. [7 marks]
\item Calculate her greatest and least speeds during her motion. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q3 [10]}}