| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Variable force (position x) - find velocity |
| Difficulty | Standard +0.3 This is a standard M3 variable force question using F=ma with v dv/dx. It requires setting up the differential equation, separating variables, and integrating x^(1/2), which are routine techniques for this module. The calculation is straightforward with clear boundary conditions and no conceptual surprises. |
| Spec | 6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(F = ma = 0.5 \frac{dv}{dx} = 3x^{\frac{1}{2}}\) | M1 | |
| \(\therefore \int v \, dv = \int 6x^{\frac{1}{4}} \, dx\) | M1 | |
| giving \(\frac{1}{2}v^2 = 4x^{\frac{5}{4}} + c\) | A1 | |
| \(x = 1, v = 2 \therefore c = -2\) | M1 | |
| \(\therefore v^2 = 8x^{\frac{5}{4}} - 4\) | A1 | |
| (b) \(x = 4\) gives \(v^2 = 64 - 4 = 60 \therefore v = \sqrt{60} = 7.7\) ms\(^{-1}\) (1dp) | M1 A1 | (7) |
(a) $F = ma = 0.5 \frac{dv}{dx} = 3x^{\frac{1}{2}}$ | M1
$\therefore \int v \, dv = \int 6x^{\frac{1}{4}} \, dx$ | M1
giving $\frac{1}{2}v^2 = 4x^{\frac{5}{4}} + c$ | A1
$x = 1, v = 2 \therefore c = -2$ | M1
$\therefore v^2 = 8x^{\frac{5}{4}} - 4$ | A1
(b) $x = 4$ gives $v^2 = 64 - 4 = 60 \therefore v = \sqrt{60} = 7.7$ ms$^{-1}$ (1dp) | M1 A1 | (7)
---2. A particle $P$ of mass 0.5 kg moves along the positive $x$-axis under the action of a single force directed away from the origin $O$. When $P$ is $x$ metres from $O$, the magnitude of the force is $3 x ^ { \frac { 1 } { 2 } } \mathrm {~N}$ and $P$ has a speed of $v \mathrm {~ms} ^ { - 1 }$.
Given that when $x = 1 , P$ is moving away from $O$ with speed $2 \mathrm {~ms} ^ { - 1 }$,
\begin{enumerate}[label=(\alph*)]
\item find an expression for $v ^ { 2 }$ in terms of $x$,
\item show that when $x = 4 , P$ has a speed of $7.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to 1 decimal place.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q2 [7]}}