| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Advanced work-energy problems |
| Type | Variable force along axis work-energy |
| Difficulty | Standard +0.3 This is a straightforward application of work-energy principles with integration. Part (a) requires integrating a simple exponential function (5 + 4e^(-x)) from 0 to 1, which is routine M3 content. Part (b) applies the work-energy theorem directly using the result from (a). Both parts follow standard textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 6.02c Work by variable force: using integration6.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| work done \(= \int_0^1 5 + 4e^{-x} \, dx = [5x - 4e^{-x}]_0^1\) | M1 A1 | |
| \(= (5 - 4e^{-1}) - (0 - 4) = 9 - 4e^{-1} = 7.53 \text{ J (3sf)}\) | M1 A1 | |
| work done \(=\) change in KE | M1 | |
| \(9 - 4e^{-1} = \frac{1}{2} \times 0.4 \times (v^2 - 2^2)\) | M1 A1 | |
| giving \(v = 6.45 \text{ ms}^{-1} \text{ (3sf)}\) | A1 | (8) |
work done $= \int_0^1 5 + 4e^{-x} \, dx = [5x - 4e^{-x}]_0^1$ | M1 A1 |
$= (5 - 4e^{-1}) - (0 - 4) = 9 - 4e^{-1} = 7.53 \text{ J (3sf)}$ | M1 A1 |
work done $=$ change in KE | M1 |
$9 - 4e^{-1} = \frac{1}{2} \times 0.4 \times (v^2 - 2^2)$ | M1 A1 |
giving $v = 6.45 \text{ ms}^{-1} \text{ (3sf)}$ | A1 | (8)2. A particle $P$ of mass 0.4 kg is moving in a straight line through a fixed point $O$.
At time $t$ seconds after it passes through $O$, the distance $O P$ is $x$ metres and the resultant force acting on $P$ is of magnitude ( $5 + 4 \mathrm { e } ^ { - x }$ ) N in the direction $O P$.
When $x = 1 , P$ is at the point $A$.
\begin{enumerate}[label=(\alph*)]
\item Find, correct to 3 significant figures, the work done in moving $P$ from $O$ to $A$.
Given that $P$ passes through $O$ with speed $2 \mathrm {~ms} ^ { - 1 }$,
\item find, correct to 3 significant figures, the speed of $P$ as it passes through $A$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q2 [8]}}