| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2009 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Force depends on time t |
| Difficulty | Standard +0.3 This is a standard M3 variable force question requiring Newton's second law (F=ma), integration of a rational function with substitution, and applying boundary conditions. The integration is straightforward once set up, and the method is a direct application of taught techniques with no novel problem-solving required. |
| Spec | 6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(3a = -\left(9 + \frac{15}{(t+1)^2}\right)\) | B1 | N2L |
| \(3v = -9t + \frac{15}{t+1} (+A)\) | M1 A1ft | Integration |
| \(v=0, t=4 \Rightarrow 0 = -36 + 3 + A \Rightarrow A = 33\) | M1 A1 | Applying initial condition |
| \(v = -3t + \frac{5}{t+1} + 11\) | ||
| \(t = 0 \Rightarrow v = 16\) | M1 A1 | (7) |
## Question 1:
| Working/Answer | Marks | Guidance |
|---|---|---|
| $3a = -\left(9 + \frac{15}{(t+1)^2}\right)$ | B1 | N2L |
| $3v = -9t + \frac{15}{t+1} (+A)$ | M1 A1ft | Integration |
| $v=0, t=4 \Rightarrow 0 = -36 + 3 + A \Rightarrow A = 33$ | M1 A1 | Applying initial condition |
| $v = -3t + \frac{5}{t+1} + 11$ | | |
| $t = 0 \Rightarrow v = 16$ | M1 A1 | **(7)** |
---\begin{enumerate}
\item A particle $P$ of mass 3 kg is moving in a straight line. At time $t$ seconds, $0 \leqslant t \leqslant 4$, the only force acting on $P$ is a resistance to motion of magnitude $\left( 9 + \frac { 15 } { ( t + 1 ) ^ { 2 } } \right) \mathrm { N }$. At time $t$ seconds the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When $t = 4 , v = 0$.
\end{enumerate}
Find the value of $v$ when $t = 0$.\\
\hfill \mbox{\textit{Edexcel M3 2009 Q1 [7]}}