| Exam Board | WJEC |
|---|---|
| Module | Unit 4 (Unit 4) |
| Year | 2018 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Particle motion - velocity/time (dv/dt = f(v,t)) |
| Difficulty | Standard +0.3 This is a standard A-level mechanics differential equation problem involving resistance proportional to velocity. Part (a) requires setting up Newton's second law with given information (routine). Part (b) involves solving a first-order linear DE using separation of variables or integrating factor (standard technique). Part (c) is a simple substitution (v=0). While it requires multiple steps and integration skills, it follows a well-established template for resistance problems with no novel insight needed, making it slightly easier than average. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)3.03d Newton's second law: 2D vectors |
An object of mass $0 \cdot 5$ kg is thrown vertically upwards with initial speed $24$ ms$^{-1}$. The velocity of the object at time $t$ seconds is $v$ ms$^{-1}$. During the upward motion, the object experiences a resistance to motion $RN$, where $R$ is proportional to $v$. When the velocity of the object is $0 \cdot 2$ ms$^{-1}$ the resistance to motion is $0 \cdot 08$ N.
\begin{enumerate}[label=(\alph*)]
\item Show that the upward motion of the object satisfies the differential equation
$$\frac{\mathrm{d}v}{\mathrm{d}t} = -9 \cdot 8 - 0 \cdot 8\,v.$$ [3]
\item Find an expression for $v$ at time $t$. [6]
\item Determine the value of $t$ when the object is at the highest point of the motion. [2]
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 4 2018 Q7 [11]}}