| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2007 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Acceleration as function of displacement (v dv/dx method) |
| Difficulty | Standard +0.8 This is a non-constant acceleration problem requiring the chain rule technique (a = v dv/dx) and integration, which is beyond basic mechanics. Students must recognize when velocity is maximum (a=0), then integrate to find that maximum velocity. While systematic, this requires understanding of calculus-based mechanics not typically seen in standard A-level, making it moderately challenging. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3 - 2x = 0 \rightarrow x = 1.5\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(v(dv/dx) = 3 - 2x\) | M1 | For using \(a = v(dv/dx)\), and attempting to solve using separation of variables |
| \(v^2/2 = 3x - x^2 \; (+C)\) | A1 | |
| \(v^2/2 = 3 \times 1.5 - 1.5^2\) | M1 | For \(C = 0\) (may be implied) and substituting ans (i) |
| Maximum value is \(2.12 \text{ ms}^{-1}\) | A1 | 4 |
## Question 2:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3 - 2x = 0 \rightarrow x = 1.5$ | B1 | |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $v(dv/dx) = 3 - 2x$ | M1 | For using $a = v(dv/dx)$, and attempting to solve using separation of variables |
| $v^2/2 = 3x - x^2 \; (+C)$ | A1 | |
| $v^2/2 = 3 \times 1.5 - 1.5^2$ | M1 | For $C = 0$ (may be implied) and substituting ans (i) |
| Maximum value is $2.12 \text{ ms}^{-1}$ | A1 | 4 | |
**Total: 5 marks**
---2 A particle starts from rest at $O$ and travels in a straight line. Its acceleration is $( 3 - 2 x ) \mathrm { ms } ^ { - 2 }$, where $x \mathrm {~m}$ is the displacement of the particle from $O$.\\
(i) Find the value of $x$ for which the velocity of the particle reaches its maximum value.\\
(ii) Find this maximum velocity.
\hfill \mbox{\textit{CAIE M2 2007 Q2 [5]}}