| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Bearing and speed from velocity vector |
| Difficulty | Moderate -0.3 This is a standard M1 kinematics question involving vector motion in 2D. Part (a) requires basic trigonometry to find a bearing, parts (b) require applying the formula r = r₀ + vt, and part (c) involves setting up and solving a simple equation when the j-components are equal. All techniques are routine for M1 with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10e Position vectors: and displacement1.10h Vectors in kinematics: uniform acceleration in vector form |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| \(\tan\theta = \frac{5}{20}\) | M1 | For \(\tan\theta = \pm\frac{5}{20}\) or \(\pm\frac{20}{5}\) or any other complete method |
| \(\theta = 14.036...°\) | A1 | For \(\pm14.04°\) or \(\pm75.96°\) |
| \(\theta = 104°\) nearest degree | A1 | For \(104°\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| \(\mathbf{p} = 400\mathbf{i} + t(15\mathbf{i} + 20\mathbf{j})\) | M1 A1 | Clear attempt at p or q; \(t\) must be attached to velocity vector; position and velocity paired correctly |
| \(\mathbf{q} = 800\mathbf{j} + t(20\mathbf{i} - 5\mathbf{j})\) | A1 | "q =" not needed but must be clear it's \(Q\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Notes |
| Equate j components: \(20t(\mathbf{j}) = (800 - 5t)(\mathbf{j})\) | M1 | For equating j components; allow j's on both sides |
| \(t = 32\) | A1 | |
| \(\mathbf{s} = 800\mathbf{j} + 32(20\mathbf{i} - 5\mathbf{j})\) | M1 | Independent mark for substituting their \(t\) into their q from (b) |
| \(= 640\mathbf{i} + 640\mathbf{j}\) | A1 |
## Question 1:
### Part (a)
| Working | Marks | Notes |
|---------|-------|-------|
| $\tan\theta = \frac{5}{20}$ | M1 | For $\tan\theta = \pm\frac{5}{20}$ or $\pm\frac{20}{5}$ or any other complete method |
| $\theta = 14.036...°$ | A1 | For $\pm14.04°$ or $\pm75.96°$ |
| $\theta = 104°$ nearest degree | A1 | For $104°$ |
### Part (b)
| Working | Marks | Notes |
|---------|-------|-------|
| $\mathbf{p} = 400\mathbf{i} + t(15\mathbf{i} + 20\mathbf{j})$ | M1 A1 | Clear attempt at **p** or **q**; $t$ must be attached to velocity vector; position and velocity paired correctly |
| $\mathbf{q} = 800\mathbf{j} + t(20\mathbf{i} - 5\mathbf{j})$ | A1 | "**q** =" not needed but must be clear it's $Q$ |
### Part (c)
| Working | Marks | Notes |
|---------|-------|-------|
| Equate **j** components: $20t(\mathbf{j}) = (800 - 5t)(\mathbf{j})$ | M1 | For equating **j** components; allow **j**'s on both sides |
| $t = 32$ | A1 | |
| $\mathbf{s} = 800\mathbf{j} + 32(20\mathbf{i} - 5\mathbf{j})$ | M1 | Independent mark for substituting their $t$ into their **q** from (b) |
| $= 640\mathbf{i} + 640\mathbf{j}$ | A1 | |
---\begin{enumerate}
\item \hspace{0pt} [In this question $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin $O$.]
\end{enumerate}
Two cars $P$ and $Q$ are moving on straight horizontal roads with constant velocities. The velocity of $P$ is $( 15 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $Q$ is $( 20 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$\\
(a) Find the direction of motion of $Q$, giving your answer as a bearing to the nearest degree.
At time $t = 0$, the position vector of $P$ is $400 \mathbf { i }$ metres and the position vector of $Q$ is 800j metres. At time $t$ seconds, the position vectors of $P$ and $Q$ are $\mathbf { p }$ metres and $\mathbf { q }$ metres respectively.\\
(b) Find an expression for\\
(i) $\mathbf { p }$ in terms of $t$,\\
(ii) $\mathbf { q }$ in terms of $t$.\\
(c) Find the position vector of $Q$ when $Q$ is due west of $P$.
\hfill \mbox{\textit{Edexcel M1 2016 Q1 [10]}}