CAIE A-Level Mathematics AS 4 Mechanics Questions

Practise Mechanics through force diagrams, equilibrium, kinematics, Newton's laws, momentum, work, energy and power using connected-particle and inclined-plane models.

Covered syllabus points

4.1 Forces and equilibrium 4.2 Kinematics of motion in a straight line 4.3 Momentum 4.4 Newton's laws of motion 4.5 Energy, work and power

[Maximum number: 6]

A block of mass 8 kg slides down a rough plane inclined at $30^{\circ}$ to the horizontal, starting from rest. The coefficient of friction between the block and the plane is $\mu$ . The block accelerates uniformly down the plane at $2.4 \mathrm{~m} \mathrm{~s}^{-2}$ .

(a)

Draw a diagram showing the forces acting on the block.

[ 1 ]

(b)

Find the value of $\mu$ .

[ 4 ]

$R=8 g \cos 30[=40 \sqrt{3}=69.282 \ldots]$
Resolving perpendicular to the plane.

Resolving parallel to the plane and attempt to apply Newton's second law.
$[8 g \sin 30-F=8 \times 2.4 \Rightarrow F=20.8]$

M1*

3 terms. Allow sign errors, sin/cos mix. Allow g
missing, otherwise dimensionally correct.

Use of $F=\mu R$ to get an equation in $\mu$ only.
$[8 g \sin 30-8 g \mu \cos 30=8 \times 2.4 \quad 40-40 \sqrt{3} \mu=19.2]$
Marking guidance:

Allow g missing in either or both of F and R.
Allow sign errors, consistent sin/cos mix.
R must be a single component of a force.
Allow the 3 masses to be cancelled.

$\mu=0.3[0 \ldots] \quad\left[\right.$ May first see $\frac{20.8}{40 \sqrt{3}}$ or $\left.\frac{20.8}{69.282 \ldots}\right]$
Allow exact value $\frac{13 \sqrt{3}}{75}$ or $\frac{104 \sqrt{3}}{600}$ oe.

(c)

Find the speed of the block after it has moved 3 m down the plane.

[ 1 ]

$\left[v^{2}=2 \times 2.4 \times 3 \Rightarrow\right.$ greatest speed $\left.=\right] 3.79 \mathrm{~ms}^{-1}=\frac{6 \sqrt{10}}{5}$
3.79473... (3.8 without a more accurate value seen gets
B0 and should be annotated SF).

[Maximum number: 5]

Figure for Question 2 — CAIE A-Level Mathematics AS

The displacement of a particle moving in a straight line is s metres at time t seconds after leaving a fixed point O. The particle starts from rest and passes through points P, Q and R, at times t=5, t=10 and t=15 respectively, and returns to O at time t=20. The distances O P, O Q and O R are 50 m , 150 m and 200 m respectively.

The diagram shows a displacement-time graph which models the motion of the particle from t=0 to t=20. The graph consists of two curved segments A B and C D and two straight line segments B C and D E.

(a)

Find the speed of the particle between t=5 and t=10.

[ 1 ]

Speed $=20 \mathrm{~ms}^{-1}$

(b)

Find the acceleration of the particle between t=0 and t=5, given that it is constant.

[ 2 ]

$20=0+a \times 5$
Use of v=u+a t OE

$a=4 \mathrm{~ms}^{-2}$

(c)

Find the average speed of the particle during its motion.

[ 2 ]

$\frac{50+100+50+200}{20}$
Use of $\frac{\text { total distance }}{\text { total time }}$ OE

Average speed $=20 \mathrm{~ms}^{-1}$

[Maximum number: 7]

A car has mass 1400 kg . When the speed of the car is $\mathrm{vms}^{-1}$ the magnitude of the resistance to motion is $k v^{2} \mathrm{~N}$ where k is a constant.

(a)

The car moves at a constant speed of $24 \mathrm{~m} \mathrm{~s}^{-1}$ up a hill inclined at an angle of $\alpha$ to the horizontal where $\sin \alpha=0.12$ . At this speed the magnitude of the resistance to motion is 480 N .

[ 4 ]

(i)

Find the value of k.

[ 1 ]

$\left[k \times 24^{2}=480 \Rightarrow\right] k=\frac{5}{6}$
OE, e.g. $\frac{480}{576}, 0.833$ or better.

(ii)

Find the power of the car's engine.

[ 3 ]

Attempt at Newton's second law
$[D F=480+1400 g \times 0.12 \Rightarrow D F=2160]$

*M1

3 terms;
allow sign errors;
allow $\sin / \cos \operatorname{mix}$ .

Marking guidance:

Allow $D F=480+1400 g \times \sin 6.9$ or better.

May see $D F=\left(\right.$ their $\left.\frac{5}{6}\right) 24^{2}+1400 g \times 0.12$ .

Power =( their 2160 $) \times 24$
For using $P=D F \times v$ , where D F is numerical.

51840 W
Allow W missing, but if given in kW units must be present.

Allow 51.84 kW .

Allow $51800,51.8 \mathrm{~kW}$ .

(b)

The car now moves at a constant speed on a straight level road.

Given that its engine is working at 54 kW , find this speed.

[ 3 ]

$D F=\frac{54000}{v}$ and $D F=\left(\right.$ their $\left.\frac{5}{6}\right) v^{2}$

*B1FT

FT their $\frac{5}{6}>0$ .

Get an equation of the form $a v^{3}=b$ and attempt to solve for v to get a
positive value
a and b must both be positive or both negative.

Must get to a value for v;
if cubic not seen, the cubic may be implied by the correct answer for their equation.

Speed $=40.2 \mathrm{~ms}^{-1}$
40.165977.

AWRT 40.2 from correct work.

[Maximum number: 9]

A car of mass 1400 kg is moving on a straight road against a constant force of 1250 N resisting the motion.

(a)

The car moves along a horizontal section of the road at a constant speed of $36 \mathrm{~m} \mathrm{~s}^{-1}$ .

[ 7 ]

(i)

Calculate the work done against the resisting force during the first 8 seconds.

[ 2 ]

[WD = $1250 \times 36 \times 8$ ]
For using Work Done = Force × Distance.
$\mathrm{WD}=360000 \mathrm{~J}$
or 360 kJ

(ii)

Calculate, in kW , the power developed by the engine of the car.

[ 2 ]

Power $=1250 \times 36$ or $\mathrm{P}=\frac{360000}{8}[=45000 \mathrm{~J}]$
B1 FT
FT Work Done from $\frac{\mathbf{a}(\mathbf{i})}{8}$ .
$=45 \mathrm{~kW}$
B1

(iii)

Given that this power is suddenly increased by 12 kW , find the instantaneous acceleration of the car.

[ 3 ]

$\mathrm{DF}=\frac{57000}{36}[=1583.3 .$ .
Use changed Power in $P=\mathrm{DF} \times v$ .
$\frac{57000}{36}-1250=1400 a$
For using Newton's 2nd law applied to the car.
$a=0.238 \mathrm{~ms}^{-2}$

(b)

The car now travels at a constant speed of $32 \mathrm{~m} \mathrm{~s}^{-1}$ up a section of the road inclined at $\theta^{\circ}$ to the horizontal, with the engine working at 64 kW .

Find the value of $\theta$ .

[ 2 ]

$\frac{64000}{32}=1250+1400 g \sin \theta$
For using DF= resistance + component of the weight of
the car.
$\theta=3.1[3.0708 \ldots]$

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question (a)

Question (b)

Question (c)

Question 3

Question (a)

Question (i)

Question (ii)

Question (b)

Question 4

Question (a)

Question (i)

Question (ii)

Question (iii)

Question (b)