| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Force depends on time t |
| Difficulty | Standard +0.3 This is a straightforward M3 kinematics question requiring integration of acceleration to find velocity, then applying initial conditions to find constants. The integration of 1/(1+t) is standard, and solving the resulting logarithmic equation in part (c) uses basic log laws. All steps are routine for M3 students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks |
|---|---|
| (a) \(\frac{dv}{dt} = \frac{-k}{1+t}\) | B1 M1 A1 |
| \(\int dv = k\int\frac{1}{1+t}dt\) | |
| \(v = k\ln(1+t) + c\) | |
| (b) \(t = 0, v = 0\), so \(c = 0\) | M1 A1 A1 |
| \(t = 2, v = 4\): \(k = \frac{4}{\ln 3}\); hence result | |
| (c) When \(v = 8\), \(8 = \frac{4}{\ln 3}\ln(1+t)\) | M1 A1 A1 |
| \(\ln(1+t) = \ln 9\) | |
| \(t = 8\) | 9 marks total |
**(a)** $\frac{dv}{dt} = \frac{-k}{1+t}$ | B1 M1 A1
$\int dv = k\int\frac{1}{1+t}dt$ |
$v = k\ln(1+t) + c$ |
**(b)** $t = 0, v = 0$, so $c = 0$ | M1 A1 A1
$t = 2, v = 4$: $k = \frac{4}{\ln 3}$; hence result |
**(c)** When $v = 8$, $8 = \frac{4}{\ln 3}\ln(1+t)$ | M1 A1 A1
$\ln(1+t) = \ln 9$ |
$t = 8$ | 9 marks totalThe acceleration $a$ ms$^{-2}$ of a particle $P$ moving in a straight line away from a fixed point $O$ is given by $a = \frac{k}{1+t}$, where $t$ is the time that has elapsed since $P$ left $O$, and $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item By solving a suitable differential equation, find an expression for the velocity $v$ ms$^{-1}$ of $P$ in terms of $t$, $k$ and another constant $c$. [3 marks]
\end{enumerate}
Given that $v = 0$ when $t = 0$ and that $v = 4$ when $t = 2$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item show that $v \ln 3 = 4 \ln (1 + t)$. [3 marks]
\item Calculate the time when $P$ has a speed of 8 ms$^{-1}$. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q4 [9]}}