| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2005 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Interception: verify/find meeting point (position vector method) |
| Difficulty | Moderate -0.8 This is a straightforward M1 mechanics question testing basic vector kinematics. Parts (a)-(c) involve routine calculations: finding speed from velocity components using Pythagoras, applying r = r₀ + vt, and solving a simple equation. Part (d) requires setting up simultaneous position equations but follows a standard interception template. The final part is a standard modelling critique. All techniques are direct applications of core M1 content with no novel problem-solving required. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10e Position vectors: and displacement3.02a Kinematics language: position, displacement, velocity, acceleration3.02d Constant acceleration: SUVAT formulae |
| Answer | Marks |
|---|---|
| 8 | (a) Speed of ball = √(52 + 82) ≈ 9.43 m s–1 |
| Answer | Marks |
|---|---|
| Do not allow on their own ‘swerve’, ‘weight of ball’. | M1 A1 |
Question 8:
8 | (a) Speed of ball = √(52 + 82) ≈ 9.43 m s–1
(b) p.v. of ball = (2i + j) + (5i + 8j)t
(c) North of B when i components same, i.e. 2 + 5t = 10
t = 1.6 s
(d) When t = 1.6, p.v. of ball = 10i + 13.8j (or j component = 13.8)
Distance travelled by 2nd player = 13.8 – 6 = 6.8
Speed = 6.8 ÷ 1.6 = 4.25 m s-1
or [(2 + 5t)i +] (1 + 8t)j = [10i +] (7 + vt)j (pv’s or j components same)
Using t = 1.6: 1 + 12.8 = 7 + 1.6v (equn in v only)
v = 4.25 m s–1
(e) Allow for friction on field (i.e. velocity of ball not constant)
or allow for vertical component of motion of ball
(a) M1 Valid attempt at speed (square, add and squ. root cpts)
(b) M1 needs non-zero p.v. + (attempt at veloc vector) x t. Must be vector
(d) 2nd M1 – allow if finding displacement vector (e.g. if using wrong time)
3rd M1 for getting speed as a scalar (and final answer must be as a scalar). But if
they get e.g. ‘4.25j’, allow M1 A0
(e) Allow ‘wind’, ‘spin’, ‘time for player to accelerate’, size of ball
Do not allow on their own ‘swerve’, ‘weight of ball’. | M1 A1
(2)
M1 A1
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(6)
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B1
(1)[In this question, the unit vectors $\mathbf{i}$ and $\mathbf{j}$ are horizontal vectors due east and north respectively.]
At time $t = 0$, a football player kicks a ball from the point $A$ with position vector $(2\mathbf{i} + \mathbf{j})$ m on a horizontal football field. The motion of the ball is modelled as that of a particle moving horizontally with constant velocity $(5\mathbf{i} + 8\mathbf{j}) \text{ m s}^{-1}$. Find
\begin{enumerate}[label=(\alph*)]
\item the speed of the ball, [2]
\item the position vector of the ball after $t$ seconds. [2]
\end{enumerate}
The point $B$ on the field has position vector $(10\mathbf{i} + 7\mathbf{j})$ m.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{2}
\item Find the time when the ball is due north of $B$. [2]
\end{enumerate}
At time $t = 0$, another player starts running due north from $B$ and moves with constant speed $v \text{ m s}^{-1}$. Given that he intercepts the ball,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{3}
\item find the value of $v$. [6]
\item State one physical factor, other than air resistance, which would be needed in a refinement of the model of the ball's motion to make the model more realistic. [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2005 Q8 [13]}}