| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Force depends on time t |
| Difficulty | Standard +0.8 This M3 variable force question requires integration of a time-dependent acceleration with careful attention to signs and initial conditions, followed by finding when velocity equals zero and integrating again for displacement. While the integration itself is straightforward, students must correctly handle the direction (negative acceleration), apply initial conditions properly, and solve a transcendental equation (3/(t+1) = 1) before a final integration. The multi-step nature and need for careful bookkeeping elevate this above average difficulty. |
| Spec | 6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{d^2x}{dt^2} = -\frac{3}{(t+1)^2}\) | M1 | |
| \(\frac{dv}{dt} = -\int 3(t+1)^{-2}dt = 3(t+1)^{-1}(+c)\) | M1 A1 | |
| \(t = 0, v = 2\): \(2 = 3 + c \Rightarrow c = -1\) | M1 | |
| \(\frac{dx}{dt} = \frac{3}{t+1} - 1\) | A1 | (5 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = \int\left[\frac{3}{t+1} - 1\right]dt = 3\ln(t+1) - t(+c')\) | M1 A1 | |
| \(t = 0, x = 0 \Rightarrow c' = 0\) | B1 | |
| \(x = 3\ln(t+1) - t\) | ||
| \(v = 0 \Rightarrow \frac{3}{t+1} = 1 \Rightarrow t = 2\) | M1 A1 | |
| \(x = 3\ln 3 - 2 = 1.295\ldots = 1.30 \text{ m (Allow 1.3)}\) | M1 A1 | (7 marks total) |
## Part (a)
$\frac{d^2x}{dt^2} = -\frac{3}{(t+1)^2}$ | M1 |
$\frac{dv}{dt} = -\int 3(t+1)^{-2}dt = 3(t+1)^{-1}(+c)$ | M1 A1 |
$t = 0, v = 2$: $2 = 3 + c \Rightarrow c = -1$ | M1 |
$\frac{dx}{dt} = \frac{3}{t+1} - 1$ | A1 | (5 marks total)
## Part (b)
$x = \int\left[\frac{3}{t+1} - 1\right]dt = 3\ln(t+1) - t(+c')$ | M1 A1 |
$t = 0, x = 0 \Rightarrow c' = 0$ | B1 |
$x = 3\ln(t+1) - t$ | |
$v = 0 \Rightarrow \frac{3}{t+1} = 1 \Rightarrow t = 2$ | M1 A1 |
$x = 3\ln 3 - 2 = 1.295\ldots = 1.30 \text{ m (Allow 1.3)}$ | M1 A1 | (7 marks total)
**[12 marks total]**At time $t = 0$, a particle $P$ is at the origin $O$ moving with speed $2$ m s$^{-1}$ along the $x$-axis in the positive $x$-direction. At time $t$ seconds $(t > 0)$, the acceleration of $P$ has magnitude $\frac{3}{(t+1)^2}$ m s$^{-2}$ and is directed towards $O$.
\begin{enumerate}[label=(\alph*)]
\item Show that at time $t$ seconds the velocity of $P$ is $\left(\frac{3}{t+1} - 1\right)$ m s$^{-1}$. [5]
\item Find, to 3 significant figures, the distance of $P$ from $O$ when $P$ is instantaneously at rest. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q6 [12]}}