| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Air resistance kv² - falling from rest or projected downward |
| Difficulty | Challenging +1.2 This is a standard M3 viscous resistance problem requiring setting up Newton's second law with a quadratic resistance term, then solving a separable differential equation using substitution and partial fractions. While it involves multiple techniques (differential equations, integration, partial fractions), these are routine M3 methods applied in a straightforward context with no novel insight required. The 13-mark allocation reflects the algebraic work rather than conceptual difficulty. |
| 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) When distance from surface \(= x\) m, \(mv\frac{dv}{dx} = mg - v^2\) | B1 M1 A1 A1 |
| (b) \(\int\frac{mv}{mg-v^2}dv = x + c\) | M1 A1 M1 A1 |
| \(\frac{1}{m}\ln(mg - v^2) = x + c\) | M1 A1 A1 |
| When \(v = 0, x = 0\): \(c = -\frac{1}{m}\ln mg\) | M1 A1 A1 |
| \(x = \frac{1}{m}\ln\frac{mv}{mg-v^2}\) | M1 A1 |
| When \(v = \sqrt{\frac{mg}{2}}\), \(x = \frac{1}{2}\ln 2\) | M1 A1 |
| Total: 13 marks |
(a) When distance from surface $= x$ m, $mv\frac{dv}{dx} = mg - v^2$ | B1 M1 A1 A1 |
(b) $\int\frac{mv}{mg-v^2}dv = x + c$ | M1 A1 M1 A1 |
$\frac{1}{m}\ln(mg - v^2) = x + c$ | M1 A1 A1 |
When $v = 0, x = 0$: $c = -\frac{1}{m}\ln mg$ | M1 A1 A1 |
$x = \frac{1}{m}\ln\frac{mv}{mg-v^2}$ | M1 A1 |
When $v = \sqrt{\frac{mg}{2}}$, $x = \frac{1}{2}\ln 2$ | M1 A1 |
| **Total: 13 marks** |A small sphere $S$, of mass $m$ kg is released from rest at the surface of a liquid in a right circular cylinder whose axis is vertical. When $S$ is moving downwards with speed $v$ ms$^{-1}$, the viscous resistive force acting upwards on it has magnitude $v^2$ N.
\begin{enumerate}[label=(\alph*)]
\item Write down a differential equation for the motion of $S$, clearly defining any symbol(s) that you introduce. [4 marks]
\item Find, in terms of $m$, the distance $S$ has fallen when its speed is $\sqrt{\frac{mg}{2}}$ ms$^{-1}$. [9 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q5 [13]}}