| Exam Board | OCR |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2016 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Collision or meeting problems |
| Difficulty | Standard +0.3 This is a multi-part kinematics question requiring integration to find velocity for particle A, reading graphs, and using kinematic equations. While it involves several steps and variable acceleration (0.18t), the techniques are standard M1 fare: integrating acceleration, using s-t relationships, and applying constant acceleration formulas. The question guides students through each stage clearly, making it slightly easier than average for M1. |
| Spec | 3.02b Kinematic graphs: displacement-time and velocity-time3.02c Interpret kinematic graphs: gradient and area3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks |
|---|---|
| Marks | Guidance |
| M1* | Integration indicated by change in coefficient and increase in power |
| A1 | |
| D*M1 | |
| A1 | |
| [4] |
| Answer | Marks |
|---|---|
| Marks | Guidance |
| M1* | Differentiation indicated by change in coefficient and reduction in power |
| D*M1 | |
| A1 | There are instances of solutions in which SB(5) =25 is used to show that \(U=3\), and then demonstrate that \(SB(5) = 25\). Such work can gain no marks. \(u = 3\) without any supporting work. M0A0. |
| A1 | |
| [4] |
| Answer | Marks |
|---|---|
| Marks | Guidance |
| M1* | Integration of v(A) |
| D*M1 | |
| A1 | Accept 123 |
| M1 | |
| A1 | |
| [5] | |
| Total Marks | 72 |
## Part i
**Answer/Working:**
A: $v=\int 0.18t$ dt
$v=0.18/2$ t$^2$ (+c)
$9=0.09t^2$
$t = 10$
| **Marks** | **Guidance** |
|-----------|-------------|
| M1* | Integration indicated by change in coefficient and increase in power |
| A1 | |
| D*M1 | |
| A1 | |
| [4] | |
## Part ii
**Answer/Working:**
B: $v = d(Ut+0.08t^2)/dt$
$v = U+0.24t^2$
$9=U+0.24x5^2$
$U = 3$
$SB(5)= 3x5+0.08x5^3$
$SB(5) = 25$ m AG
| **Marks** | **Guidance** |
|-----------|-------------|
| M1* | Differentiation indicated by change in coefficient and reduction in power |
| D*M1 | |
| A1 | There are instances of solutions in which SB(5) =25 is used to show that $U=3$, and then demonstrate that $SB(5) = 25$. Such work can gain no marks. $u = 3$ without any supporting work. M0A0. |
| A1 | |
| [4] | |
## Part iii
**Answer/Working:**
A: $x=\int 0.09t^2dt$
$x=0.09t^3 /3$
$x(16)=0.03x16^3$
$x=122.88$ (may be implied by later work)
$122.88-25+10x9+(9+v)(x1)/2$
$v=6.76$ m s$^{-1}$
OR
$122.88-25-10x9 =9x1+/-ax1^2 /2$
Deceleration = 2.24 m s$^{-2}$
$v = 9 - 2.24x1$
$v = 6.76$ m s$^{-1}$
| **Marks** | **Guidance** |
|-----------|-------------|
| M1* | Integration of v(A) |
| D*M1 | |
| A1 | Accept 123 |
| M1 | |
| A1 | |
| [5] | |
| **Total Marks** | **72** |
|---|---|
---
**General Guidance:** $s = ut +/- a t^2 / 2$\includegraphics{figure_7}
The diagram shows the $(t, v)$ graphs for two particles $A$ and $B$ which move on the same straight line. The units of $v$ and $t$ are $\text{m s}^{-1}$ and $\text{s}$ respectively. Both particles are at the point $S$ on the line when $t = 0$. The particle $A$ is initially at rest, and moves with acceleration $0.18t\text{ m s}^{-2}$ until the two particles collide when $t = 16$. The initial velocity of $B$ is $U\text{ m s}^{-1}$ and $B$ has variable acceleration for the first five seconds of its motion. For the next ten seconds of its motion $B$ has a constant velocity of $9\text{ m s}^{-1}$; finally $B$ moves with constant deceleration for one second before it collides with $A$.
\begin{enumerate}[label=(\roman*)]
\item Calculate the value of $t$ at which the two particles have the same velocity. [4]
\end{enumerate}
For $0 \leq t \leq 5$ the distance of $B$ from $S$ is $(Ut + 0.08t^2)\text{ m}$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Calculate $U$ and verify that when $t = 5$, $B$ is $25\text{ m}$ from $S$. [4]
\item Calculate the velocity of $B$ when $t = 16$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR M1 2016 Q7 [13]}}