| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2014 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Variable acceleration with initial conditions |
| Difficulty | Moderate -0.3 This is a standard M2 variable acceleration question requiring integration of acceleration to find velocity (with initial conditions), solving a quadratic to find when v=0, then integrating velocity to find displacement. All steps are routine textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 3.02a Kinematics language: position, displacement, velocity, acceleration3.02d Constant acceleration: SUVAT formulae3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(v = \int a\,dt = \int 2t - 3\,dt = t^2 - 3t (+C)\) | M1A1 | M1 for attempt to integrate (one power increasing by 1); A1 for correct integral without \(c\) |
| \(t=0, v=2 \Rightarrow C=2\), \(v = t^2 - 3t + 2\) | M1A1 | M1 for using \(t=0\), \(v=2\) to find \(c\); A1 for answer ('\(v=\)' not needed) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(v = 0\) | M1 | First M1 for setting their \(v\) expression equal to zero |
| \((t-1)(t-2) = 0\) | M1 | M1 for solving for \(t\) (must be a quadratic); mark implied by two correct answers |
| \(t_1 = 1\), \(t_2 = 2\) | A1 | A1 for both answers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(s = \int_1^2 t^2 - 3t + 2\,dt = \left[\frac{1}{3}t^3 - \frac{3}{2}t^2 + 2t\right]_1^2\) | M1A1 | M1 for attempt to integrate their \(v\) (all powers increasing by 1); A1 for correct integral (NOT ft), without \(c\) |
| \(= \left(\frac{8}{3} - 6 + 4\right) - \left(\frac{1}{3} - \frac{3}{2} + 2\right)\) or \(= \frac{1}{3}(8-1) - \frac{3}{2}(4-1) + 2(2-1)\) | DM1 | Dependent on first M1; substituting their \(t\) values and subtracting (either way round) |
| Distance \(= \frac{1}{6}\), 0.17 or better (m) | A1 | Must be positive; N.B. if they add or subtract some other distance it's M0 |
# Question 1:
## Part 1a:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $v = \int a\,dt = \int 2t - 3\,dt = t^2 - 3t (+C)$ | M1A1 | M1 for attempt to integrate (one power increasing by 1); A1 for correct integral without $c$ |
| $t=0, v=2 \Rightarrow C=2$, $v = t^2 - 3t + 2$ | M1A1 | M1 for using $t=0$, $v=2$ to find $c$; A1 for answer ('$v=$' not needed) |
## Part 1b:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $v = 0$ | M1 | First M1 for setting their $v$ expression equal to zero |
| $(t-1)(t-2) = 0$ | M1 | M1 for solving for $t$ (must be a quadratic); mark implied by two correct answers |
| $t_1 = 1$, $t_2 = 2$ | A1 | A1 for both answers |
## Part 1c:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $s = \int_1^2 t^2 - 3t + 2\,dt = \left[\frac{1}{3}t^3 - \frac{3}{2}t^2 + 2t\right]_1^2$ | M1A1 | M1 for attempt to integrate their $v$ (all powers increasing by 1); A1 for correct integral (NOT ft), without $c$ |
| $= \left(\frac{8}{3} - 6 + 4\right) - \left(\frac{1}{3} - \frac{3}{2} + 2\right)$ or $= \frac{1}{3}(8-1) - \frac{3}{2}(4-1) + 2(2-1)$ | DM1 | Dependent on first M1; substituting their $t$ values and subtracting (either way round) |
| Distance $= \frac{1}{6}$, 0.17 or better (m) | A1 | Must be positive; N.B. if they add or subtract some other distance it's M0 |
---\begin{enumerate}
\item A particle $P$ moves on the $x$-axis. The acceleration of $P$, in the positive $x$ direction at time $t$ seconds, is $( 2 t - 3 ) \mathrm { m } \mathrm { s } ^ { - 2 }$. The velocity of $P$, in the positive $x$ direction at time $t$ seconds, is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When $t = 0 , v = 2$\\
(a) Find $v$ in terms of $t$.
\end{enumerate}
The particle is instantaneously at rest at times $t _ { 1 }$ seconds and $t _ { 2 }$ seconds, where $t _ { 1 } < t _ { 2 }$.\\
(b) Find the values $t _ { 1 }$ and $t _ { 2 }$.\\
(c) Find the distance travelled by $P$ between $t = t _ { 1 }$ and $t = t _ { 2 }$.\\
\hfill \mbox{\textit{Edexcel M2 2014 Q1 [11]}}