| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Acceleration from velocity differentiation |
| Difficulty | Standard +0.3 This is a straightforward vector mechanics question requiring differentiation of logarithmic and exponential functions to find acceleration, then applying F=ma. The techniques are standard M3 content with no problem-solving insight needed—just direct application of calculus and Newton's second law. Slightly easier than average due to its routine nature. |
| Spec | 1.06b Gradient of e^(kx): derivative and exponential model1.06d Natural logarithm: ln(x) function and properties1.10h Vectors in kinematics: uniform acceleration in vector form3.03d Newton's second law: 2D vectors |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(a = \frac{d}{dt}(v) = (-\frac{1}{t+1} + 2e^{-2t}\mathbf{j})\) ms\(^{-2}\) | M1 A2 | |
| (b) \(t = 1\), \(a = \frac{1}{2}\mathbf{i} + 2e^{-2}\mathbf{j}\) | M1 | |
| \( | a | = \sqrt{\frac{1}{4} + 4e^{-4}} = 0.5686\) |
| \(F = ma = 0.25 \times 0.5686 = 0.142\) N (3sf) | A1 | (7) |
(a) $a = \frac{d}{dt}(v) = (-\frac{1}{t+1} + 2e^{-2t}\mathbf{j})$ ms$^{-2}$ | M1 A2 |
(b) $t = 1$, $a = \frac{1}{2}\mathbf{i} + 2e^{-2}\mathbf{j}$ | M1 |
$|a| = \sqrt{\frac{1}{4} + 4e^{-4}} = 0.5686$ | M1 A1 |
$F = ma = 0.25 \times 0.5686 = 0.142$ N (3sf) | A1 | (7)2. A particle $P$ of mass 0.25 kg is moving on a horizontal plane.
At time $t$ seconds the velocity, $\mathbf { v } \mathrm { ms } ^ { - 1 }$, of $P$ relative to a fixed origin $O$ is given by
$$\mathbf { v } = \ln ( t + 1 ) \mathbf { i } - \mathrm { e } ^ { - 2 t } \mathbf { j } , t \leq 0 ,$$
where $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors in the horizontal plane.
\begin{enumerate}[label=(\alph*)]
\item Find the acceleration of $P$ in terms of $t$.
\item Find, correct to 3 significant figures, the magnitude of the resultant force acting on $P$ when $t = 1$.\\
(4 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q2 [7]}}