| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2004 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Time to reach given speed |
| Difficulty | Standard +0.8 This is a variable force mechanics problem requiring Newton's second law to derive a differential equation (part i is routine), then separating variables and integrating a non-trivial rational function with partial fractions (part ii). The integration requires factorizing 7000-v², decomposing into partial fractions, and evaluating definite integrals with logarithms—significantly more demanding than standard constant acceleration problems but still within M2 scope. |
| Spec | 6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| (i) For using Newton's second law and \(a = \frac{dv}{dt}\) | M1 | |
| Correct working to obtain the given answer | A1 | 2 marks |
| (ii) For separating the variables and attempting to integrate | M1 | |
| \(t = -125\ln(7000 - v^2)\) \((+C)\) (aef) | A1 | |
| For attempting to find \(t(40) - t(10)\) (or equivalent) | M1 | |
| Time taken is 30.6 s | A1 | 4 marks |
**(i)** For using Newton's second law and $a = \frac{dv}{dt}$ | M1 |
Correct working to obtain the given answer | A1 | 2 marks
**(ii)** For separating the variables and attempting to integrate | M1 |
$t = -125\ln(7000 - v^2)$ $(+C)$ (aef) | A1 |
For attempting to find $t(40) - t(10)$ (or equivalent) | M1 |
Time taken is 30.6 s | A1 | 4 marks
---3 A car of mass 1000 kg is moving on a straight horizontal road. The driving force of the car is $\frac { 28000 } { v } \mathrm {~N}$ and the resistance to motion is $4 \nu \mathrm {~N}$, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the speed of the car $t$ seconds after it passes a fixed point on the road.\\
(i) Show that $\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 7000 - v ^ { 2 } } { 250 v }$.
The car passes points $A$ and $B$ with speeds $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively.\\
(ii) Find the time taken for the car to travel from $A$ to $B$.
\hfill \mbox{\textit{CAIE M2 2004 Q3 [6]}}