Edexcel A-Level Mathematics AS M1.2 Vectors in Mechanics Questions

Practise using vector components, magnitudes, directions and position vectors to model forces, velocities, bearings and motion in a plane.

Exam points

resolve vectors into i and j components before finding resultants or magnitudes
use position and velocity vectors to find speeds, bearings and relative motion
form equations from parallel vectors, equal speeds or shared directions

Covered syllabus points

M1.2.1 - Magnitude and direction M1.2.2 - Application of vectors

[Maximum number: 4]

Figure 4

A particle P of mass 4 m lies on the surface of a fixed rough inclined plane.
The plane is inclined to the horizontal at an angle $\alpha$ where $\tan \alpha=\frac{3}{4}$
The particle P is attached to one end of a light inextensible string.
The string passes over a small smooth pulley that is fixed at the top of the plane. The other end of the string is attached to a particle Q of mass m which lies on a smooth horizontal plane.

The string lies along the horizontal plane and in the vertical plane that contains the pulley and a line of greatest slope of the inclined plane.

The system is released from rest with the string taut, as shown in Figure 4, and P moves down the plane.

The coefficient of friction between P and the plane is $\frac{1}{4}$
For the motion before Q reaches the pulley

find the magnitude of the force exerted on the pulley by the string.

$2 T \sin \frac{1}{2} \alpha$ oe e.g. $\sqrt{T^{2}+T^{2}-2 T^{2} \cos \alpha}$ using cos rule

Or $\frac{T \sin \alpha}{\sin \left(90^{\circ}-\frac{1}{2} \alpha\right)}$ using sine rule

Or $\sqrt{(T-T \cos \alpha)^{2}+(T \sin \alpha)^{2}}$ using components and Pythag.
Substitute for T and trig
$\frac{8 m g \sqrt{10}}{125}$ oe, 2 m or 2.0 m or 1.98 m or 0.2 m g or better
(4)

[Maximum number: 10]

[In this question i and j are horizontal unit vectors directed due east and due north respectively.]

A particle P moves with constant acceleration $(-\lambda \mathbf{i}+2 \lambda \mathbf{j}) \mathrm{ms}^{-2}$ , where $\lambda$ is a positive constant.

At time t=0, the velocity of P is $(5 \mathbf{i}-8 \mathbf{j}) \mathrm{m} \mathrm{s}^{-1}$

(a)

Find the velocity of P when $t=5 \mathrm{~s}$ , giving your answer in terms of i, j and $\lambda$ .

The speed of P when $t=5 \mathrm{~s}$ is $13 \mathrm{~ms}^{-1}$

[ 2 ]

$(5 \mathbf{i}-8 \mathbf{j})+5(-\lambda \mathbf{i}+2 \lambda \mathbf{j})\left(\mathrm{m} \mathrm{s}^{-1}\right)$ isw
(2)

(b)

Show that

25 \lambda^{2}-42 \lambda-16=0

[ 3 ]

$13=\sqrt{(5-5 \lambda)^{2}+(-8+10 \lambda)^{2}}$

M1 A1

$169=25-50 \lambda+25 \lambda^{2}+64-160 \lambda+100 \lambda^{2}$ $25 \lambda^{2}-42 \lambda-16=0^{*}$

A1* cso

(3)

(c)

Find the direction of motion of P when $t=4 \mathrm{~s}$ , giving your answer as a bearing to the nearest degree.

[ 5 ]

( -2 i+4 j ) seen or implied
(5 i-8 j)+(-2 i+4 j) 4

Figure for Answer to Question (c) — Edexcel A-Level Mathematics AS

$339^{\circ}$
(5)

(10)

Notes for question 4

(a)
Or column vector

Complete method to find the velocity when t=4.

Correct unsimplified expression. Note the correct velocity is v=-3 i+8 j

Use their velocity vector at t=4 with trig to find a relevant angle.

Cao. Degrees sign not required.

N.B. if they work with both values of $\lambda$ , can score max all the marks except the last one.

QUESTION
NUMBER

SCHEME

MARKS

5

[Maximum number: 4]

Figure 4

One end of a light inextensible string is attached to a particle A of mass 2 m. The other end of the string is attached to a particle B of mass 3 m. The string passes over a small, smooth, light pulley P which is fixed at the top of a rough inclined plane. The plane is inclined to the horizontal at an angle $\alpha$ , where $\tan \alpha=\frac{3}{4}$

Particle A is held at rest on the plane with the string taut and B hanging freely below P, as shown in Figure 4. The section of the string AP is parallel to a line of greatest slope of the plane.

The coefficient of friction between A and the plane is $\frac{1}{2}$
Particle A is released and begins to move up the plane.
For the motion before A reaches the pulley,

find the magnitude of the force exerted on the pulley by the string.

$T=\frac{12 m g}{5}(23.52 m)$ $2 T \cos \left(\frac{90^{\circ}-\alpha}{2}\right) \quad$ OR $\quad \sqrt{T^{2}+T^{2}-2 T^{2} \cos \left(90^{\circ}+\alpha\right)} \quad$ OR
$\sqrt{(T \cos \alpha)^{2}+(T+T \sin \alpha)^{2}}$
Substitute for trig and T to obtain an expression in m or m g
$\frac{48 \sqrt{5} m g}{25}$ ;
Accept 4.3 m g or better, 42 m or 42.1 m
(4)

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question (c)

Question 3