| Exam Board | AQA |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Acceleration as function of velocity (separation of variables) |
| Difficulty | Standard +0.8 This M2 question requires recognizing that a = dv/dt = -λ/v^(1/4), separating variables to integrate v^(1/4)dv, and rearranging to make v the subject involving a fifth power. While the integration itself is routine, the fractional power and the algebraic manipulation to isolate v (requiring raising to the 4/5 power) elevate this beyond a standard M2 question, making it moderately challenging but still within expected M2 scope. |
| Spec | 6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dv}{dt} = -\frac{\lambda}{v^{\frac{1}{4}}}\) | B1 | Correct differential equation |
| \(v^{\frac{1}{4}}\,dv = -\lambda\,dt\) | M1 | Separate variables |
| \(\int v^{\frac{1}{4}}\,dv = -\lambda\int dt\) | M1 | Attempt integration both sides |
| \(\frac{v^{\frac{5}{4}}}{\frac{5}{4}} = -\lambda t + c\) | A1 | Correct integration |
| \(\frac{4}{5}v^{\frac{5}{4}} = -\lambda t + c\) | ||
| When \(t=0\), \(v=u\): \(c = \frac{4}{5}u^{\frac{5}{4}}\) | M1 | Apply initial condition |
| \(\frac{4}{5}v^{\frac{5}{4}} = -\lambda t + \frac{4}{5}u^{\frac{5}{4}}\) | A1 | |
| \(v^{\frac{5}{4}} = u^{\frac{5}{4}} - \frac{5\lambda t}{4}\) | ||
| \(v = \left(u^{\frac{5}{4}} - \frac{5\lambda t}{4}\right)^{\frac{4}{5}}\) | A1 |
# Question 5:
| $\frac{dv}{dt} = -\frac{\lambda}{v^{\frac{1}{4}}}$ | B1 | Correct differential equation |
|---|---|---|
| $v^{\frac{1}{4}}\,dv = -\lambda\,dt$ | M1 | Separate variables |
| $\int v^{\frac{1}{4}}\,dv = -\lambda\int dt$ | M1 | Attempt integration both sides |
| $\frac{v^{\frac{5}{4}}}{\frac{5}{4}} = -\lambda t + c$ | A1 | Correct integration |
| $\frac{4}{5}v^{\frac{5}{4}} = -\lambda t + c$ | | |
| When $t=0$, $v=u$: $c = \frac{4}{5}u^{\frac{5}{4}}$ | M1 | Apply initial condition |
| $\frac{4}{5}v^{\frac{5}{4}} = -\lambda t + \frac{4}{5}u^{\frac{5}{4}}$ | A1 | |
| $v^{\frac{5}{4}} = u^{\frac{5}{4}} - \frac{5\lambda t}{4}$ | | |
| $v = \left(u^{\frac{5}{4}} - \frac{5\lambda t}{4}\right)^{\frac{4}{5}}$ | A1 | |5 A particle is moving along a straight line. At time $t$, the velocity of the particle is $v$. The acceleration of the particle throughout the motion is $- \frac { \lambda } { v ^ { \frac { 1 } { 4 } } }$, where $\lambda$ is a positive constant. The velocity of the particle is $u$ when $t = 0$.
Find $v$ in terms of $u , \lambda$ and $t$.\\
(7 marks)
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\hfill \mbox{\textit{AQA M2 2010 Q5 [7]}}