| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2016 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Resultant of two vector forces (direction/magnitude conditions) |
| Difficulty | Moderate -0.3 This is a straightforward M1 mechanics question testing standard vector methods. Part (a) requires setting up F₂ as a scalar multiple of a direction vector and using the resultant condition—routine algebraic manipulation. Part (b) is direct application of constant acceleration formulas with vectors. Both parts are textbook-standard with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10c Magnitude and direction: of vectors3.02f Non-uniform acceleration: using differentiation and integration3.03a Force: vector nature and diagrams3.03d Newton's second law: 2D vectors3.03p Resultant forces: using vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{F}_2 = k\mathbf{i} + k\mathbf{j}\) | B1 | \(k \neq 1\); seen or implied in working |
| \((-1+a)\mathbf{i} + (2+b)\mathbf{j}\) | M1 | Adding 2 forces with \(\mathbf{i}\)'s and \(\mathbf{j}\)'s collected; M0 if \(a\) and \(b\) both assumed to be 1 |
| \(\dfrac{-1+a}{2+b} = \dfrac{1}{3}\) | DM1 A1 | Dependent on first M1; ratio of components \(= \frac{1}{3}\) or \(\frac{3}{1}\) |
| \(a = b = k = 2.5\); \(\mathbf{F}_2 = 2.5\mathbf{i} + 2.5\mathbf{j}\) | DM1 A1; A1 | Dependent on previous M marks; solving for \(k\); correct \(k\) value; correct \(\mathbf{F}_2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{v} = 3\mathbf{i} - 22\mathbf{j} + 3(3\mathbf{i} + 9\mathbf{j})\) | M1 | |
| \(= 12\mathbf{i} + 5\mathbf{j}\) | A1 | |
| \( | \mathbf{v} | = \sqrt{12^2 + 5^2} = 13 \text{ ms}^{-1}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Use \(\mathbf{v} = \mathbf{u} + \mathbf{a}t\) with \(t = 3\) | M1 | First M1 for use of formula |
| \(12\mathbf{i} + 5\mathbf{j}\) | A1 | First A1; if wrong v seen, award A0 |
| Finding magnitude of their v | M1 | Second M1 |
| \(13\) | A1 | Second A1 |
## Question 7(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{F}_2 = k\mathbf{i} + k\mathbf{j}$ | B1 | $k \neq 1$; seen or implied in working |
| $(-1+a)\mathbf{i} + (2+b)\mathbf{j}$ | M1 | Adding 2 forces with $\mathbf{i}$'s and $\mathbf{j}$'s collected; M0 if $a$ and $b$ both assumed to be 1 |
| $\dfrac{-1+a}{2+b} = \dfrac{1}{3}$ | DM1 A1 | Dependent on first M1; ratio of components $= \frac{1}{3}$ or $\frac{3}{1}$ |
| $a = b = k = 2.5$; $\mathbf{F}_2 = 2.5\mathbf{i} + 2.5\mathbf{j}$ | DM1 A1; A1 | Dependent on previous M marks; solving for $k$; correct $k$ value; correct $\mathbf{F}_2$ |
## Question 7(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{v} = 3\mathbf{i} - 22\mathbf{j} + 3(3\mathbf{i} + 9\mathbf{j})$ | M1 | |
| $= 12\mathbf{i} + 5\mathbf{j}$ | A1 | |
| $|\mathbf{v}| = \sqrt{12^2 + 5^2} = 13 \text{ ms}^{-1}$ | M1 A1 | cso |
## Question (b) [Velocity/Kinematics]:
| Working/Answer | Marks | Guidance |
|---|---|---|
| Use $\mathbf{v} = \mathbf{u} + \mathbf{a}t$ with $t = 3$ | M1 | First M1 for use of formula |
| $12\mathbf{i} + 5\mathbf{j}$ | A1 | First A1; if wrong **v** seen, award A0 |
| Finding magnitude of their **v** | M1 | Second M1 |
| $13$ | A1 | Second A1 |
---7. Two forces $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ act on a particle $P$.
The force $\mathbf { F } _ { 1 }$ is given by $\mathbf { F } _ { 1 } = ( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }$ and $\mathbf { F } _ { 2 }$ acts in the direction of the vector $( \mathbf { i } + \mathbf { j } )$.\\
Given that the resultant of $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 2 }$ acts in the direction of the vector ( $\mathbf { i } + 3 \mathbf { j }$ ),
\begin{enumerate}[label=(\alph*)]
\item find $\mathbf { F } _ { 2 }$\\
(7)
The acceleration of $P$ is $( 3 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$. At time $t = 0$, the velocity of $P$ is $( 3 \mathbf { i } - 22 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$
\item Find the speed of $P$ when $t = 3$ seconds.\\
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2016 Q7 [11]}}