A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 12: Vectors · Lesson 3

Vectors in mechanics

118 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-vectors-in-mechanics/

Date

Do Now

Quick questions to start.

A1
Needed today
Find the unit vector in the direction of $2 i - 6 j + 3 k$ .
[2 marks]
Answer
A2
From Year 1
A particle of mass $2$ kg is acted on by a single force $(6 i - 8 j)$ N. Find its acceleration vector and the magnitude of its acceleration.
[2 marks]
Answer
A3
Last lesson
Show that the points $A (1, 4, - 2)$ , $B (4, 3, 0)$ and $C (10, 1, 4)$ are collinear.
[2 marks]
Answer
A4
Common slip
A particle has velocity $(3 i - 4 j + 12 k)$ m s $^{- 1}$ . Find its speed.
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Three forces $(2 i + j - 3 k)$ N, $(i - 4 j + 5 k)$ N and $(3 i + 3 j + k)$ N act on a particle. Find the resultant force and its magnitude.

Hint Add the

i

parts, then the

j

parts, then the

k

parts. The magnitude is found from the resultant, not by adding the three magnitudes.

B1
Three forces $(3 i - 2 j + 4 k)$ N, $(- i + 5 j + 2 k)$ N and $(4 i + j - 3 k)$ N act on a particle. Find the magnitude of the resultant force and the angle it makes with $k$ .
[4 marks]
Answer

Example 2

The forces $(a i + 2 j - k)$ N, $(3 i + b j + 4 k)$ N and $(- 5 i + j + c k)$ N act on a particle in equilibrium. Find $a$ , $b$ and $c$ .

Hint In equilibrium the resultant is the zero vector, so the

i

parts add to

0

, the

j

parts add to

0

and the

k

parts add to

0

B2
The forces $F_{1} = (2 i + p j - 3 k)$ N, $F_{2} = (- 5 i + 2 j + q k)$ N, $F_{3} = (r i - 6 j + 4 k)$ N and $F_{4} = (i + j + k)$ N act on a particle in equilibrium. Find $p$ , $q$ and $r$ . The force $F_{4}$ is then removed. Find the magnitude of the resultant of the other three.
[4 marks]
Answer

Example 3

A particle of mass $2$ kg is acted on by the forces $(3 i + j - 2 k)$ N and $(5 i - 3 j + 4 k)$ N only. Find its acceleration and the magnitude of the acceleration.

Hint

F = m a

works on the whole vector: find the resultant force, then divide every component by the mass.

B3
A particle of mass $3$ kg is acted on by the forces $(4 i - 5 j + 7 k)$ N and $(2 i + 11 j - k)$ N only. Find its acceleration, the magnitude of the acceleration, and the angle the acceleration makes with $j$ .
[5 marks]
Answer

Example 4

A drone starts at the point with position vector $(2 i - j + 4 k)$ m and moves with constant velocity $(3 i + 2 j - k)$ m s $^{- 1}$ . Find its position after $4$ s and its distance from $O$ then.

Hint At constant velocity,

r = r_{0} + v t

B4
A drone starts at $(5 i + 2 j + k)$ m and moves with constant velocity $(- 2 i + 4 j + 3 k)$ m s $^{- 1}$ . $k$ is vertically up. Find its position after $2$ s, the time at which it is $13$ m above the ground, and the distance it has travelled by then.
[5 marks]
Answer

Example 5

A particle has initial velocity $(2 i - 3 j + k)$ m s $^{- 1}$ and constant acceleration $(i + 2 j - k)$ m s $^{- 2}$ . Find its velocity and speed after $3$ s.

Hint The constant acceleration formulae hold with vectors:

v = u + a t

and

s = u t + \frac{1}{2} a t^{2}

B5
A particle has initial velocity $(4 i + j - 2 k)$ m s $^{- 1}$ and constant acceleration $(- 2 i + 3 j + k)$ m s $^{- 2}$ . Find the time at which it moves parallel to $j$ , its speed then, and its displacement from its start at that time.
[6 marks]
Answer

Practice

A-Level

C1
Two forces $(2 i + 3 j + 6 k)$ N and $(4 i - 4 j - 7 k)$ N act on a particle. Leo says the magnitudes are $7$ N and $9$ N, so the resultant has magnitude $16$ N. Explain his mistake and find the magnitude of the resultant.
[4 marks]
Answer

C2
A particle of mass $2$ kg moves under gravity and a force $(4 i - 2 j + 30 k)$ N only, where $k$ is vertically up. Take $g = 9.8$ m s $^{- 2}$ . Sam writes $a = 2 i - j + 15 k$ . Find his mistake and the magnitude of the acceleration.
[5 marks]
Answer
C3
A particle is in equilibrium under three forces. $F_{1} = (3 i - 2 j + k)$ N and $F_{2} = (- i + 5 j + 2 k)$ N. Ana writes $F_{3} = F_{1} + F_{2} = 2 i + 3 j + 3 k$ . Correct her answer and find the exact magnitude of $F_{3}$ .
[4 marks]
Answer
C4
The force $F = (p i + 2 j - 6 k)$ N has magnitude $9$ N, and $p > 0$ . Find $p$ and the angle $F$ makes with $k$ . Explain why the answer is not $48. 2^{\circ}$ .
[5 marks]
Answer
C5
A boat moves with constant velocity. At $t = 0$ it is at $A (2, - 3, 5)$ and at $t = 4$ s it is at $B (14, 1, - 1)$ , in metres. Find its velocity, its speed, and its position at $t = 6$ s.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
Two particles $P$ and $Q$ move with constant velocities. At $t = 0$ , $P$ is at $(i + 7 j + 2 k)$ m and $Q$ is at $(13 i + j + 8 k)$ m. $P$ moves with velocity $(3 i - j + 2 k)$ m s $^{- 1}$ and $Q$ with velocity $(- i + j)$ m s $^{- 1}$ .
(a)
Write down the position vectors of $P$ and $Q$ at time $t$ seconds.
[2 marks]
Answer
(b)
Show that the particles collide, and find the position vector of the point $C$ where they collide.
[4 marks]
Answer
(c)
Find the distance between the particles at $t = 1$ .
[2 marks]
Answer
(d)
Find the exact distance of $C$ from $O$ .
[2 marks]
Answer
Total for D1: 10 marks

D2
A particle of mass $2$ kg moves under a single constant force. Its velocity is $(i + 2 j + 2 k)$ m s $^{- 1}$ at $t = 0$ and $(9 i + 6 j - 6 k)$ m s $^{- 1}$ at $t = 4$ s. Jo finds the speeds, $3$ and $153$ , and says the force has magnitude $2 \times \frac{153 - 3}{4} \approx 4.7$ N.
Jo is wrong.
(a)
Find the acceleration of the particle.
[3 marks]
Answer
(b)
Find the magnitude of the force.
[2 marks]
Answer
(c)
Explain Jo's mistake.
[2 marks]
Answer
(d)
Find the speed of the particle at $t = 6$ s.
[2 marks]
Answer
Total for D2: 9 marks
D3
A particle of mass $4$ kg is released from rest at $O$ . It moves under gravity and two constant forces $(6 i - 3 j + 40 k)$ N and $(2 i + 7 j + 7.2 k)$ N, where $k$ is vertically up and $g = 9.8$ m s $^{- 2}$ . Find the distance of the particle from $O$ after $5$ s, and its speed at that moment.
[7 marks]
Answer
D4
A particle has initial velocity $(- 5 i + 2 j - 3 k)$ m s $^{- 1}$ and constant acceleration $(2 i - j + 2 k)$ m s $^{- 2}$ .
(a)
Find the velocity of the particle at time $t$ seconds.
[2 marks]
Answer
(b)
Show that the square of the speed at time $t$ is $9 t^{2} - 36 t + 38$ .
[3 marks]
Answer
(c)
Hence find the least speed of the particle and the time at which it happens.
[3 marks]
Answer
(d)
Find the times at which the speed is $11$ m s $^{- 1}$ .
[2 marks]
Answer
Total for D4: 10 marks
D5
A particle of mass $m$ kg hangs in equilibrium under gravity and the forces $F_{1} = (4 i - 2 j + 20 k)$ N, $F_{2} = (- i + 6 j + 15 k)$ N and $F_{3} = (p i + q j + 14 k)$ N. $k$ is vertically up and $g = 9.8$ m s $^{- 2}$ .
(a)
Find $p$ and $q$ .
[2 marks]
Answer
(b)
Find $m$ .
[2 marks]
Answer
(c)
The force $F_{3}$ is removed. Find the acceleration of the particle and its magnitude.
[3 marks]
Answer
(d)
Find the angle between this acceleration and the downward vertical.
[2 marks]
Answer
Total for D5: 9 marks
D6
A ball is projected from $O$ on horizontal ground with velocity $(6 i + 8 j + 19.6 k)$ m s $^{- 1}$ , where $k$ is vertically up. It is modelled as a particle moving under gravity only, with $g = 9.8$ m s $^{- 2}$ .
(a)
Find the position vector of the ball at time $t$ seconds.
[2 marks]
Answer
(b)
Find the time the ball lands and its distance from $O$ when it lands.
[4 marks]
Answer
(c)
Find the greatest height of the ball, and state one way in which the model could be improved.
[3 marks]
Answer
Total for D6: 9 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Two drones $A$ and $B$ move with constant velocities. At time $t$ seconds their position vectors, in metres, are $a = (i + 2 j + 3 k) + t (2 i - j + k)$ and $b = (7 i + 9 k) + t (- i + j - k)$ . Show that the drones do not collide. Find the least distance between them and where each drone is at that moment.
[9 marks]
Hint 1 · What to try
Write $A B = b - a$ in terms of $t$ .
Hint 2 · The first line
The square of the distance is a quadratic in $t$ . Complete the square.
Hint 3 · The full method
If the least value of the square of the distance is not zero, they never meet.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the resultant of several forces and the unknown forces that keep a particle in equilibrium.
I can use F = ma with force and acceleration vectors, including the weight of the particle.
I can use r = r0 + vt and the constant acceleration formulae in vector form.
I can find a speed, an angle with an axis, and the least distance between two moving particles.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers