| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2013 |
| Session | November |
| Topic | Variable Force |
| Type | Air resistance kv² - projected vertically upward |
| Difficulty | Challenging +1.2 This is a standard variable force mechanics problem requiring Newton's second law with air resistance, followed by separation of variables and integration. Part (i) is routine equation setup (showing a given result), while part (ii) involves standard integration techniques with inverse tan and logarithms. The calculations are somewhat involved but follow well-established methods taught in Further Maths mechanics courses, making it moderately above average difficulty. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)6.06a Variable force: dv/dt or v*dv/dx methods |
12 A bullet of mass 0.0025 kg is fired vertically upwards from a point $O$. At time $t \mathrm {~s}$ after projection the speed of the bullet is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and the resistance to motion has magnitude $0.00001 v ^ { 2 } \mathrm {~N}$.\\
(i) Show that, while the bullet is rising,
$$250 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - 2500 - v ^ { 2 }$$
(ii) It is given that the speed of projection is $350 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find
\begin{enumerate}[label=(\alph*)]
\item the time taken after projection for the bullet to reach its greatest height above $O$,
\item the greatest height above $O$ reached by the bullet.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2013 Q12}}