CAIE A-Level Mathematics 4 Mechanics Question Bank

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

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: 4]

A crate is being pushed in a straight line along a horizontal surface by a force of magnitude 25 N inclined at $20^{\circ}$ above the horizontal. The crate moves a distance of 12 m in 8 seconds with constant speed.

(a)

Find the constant speed of the crate.

[ 1 ]

(b)

Find the work done by the 25 N force.

[ 2 ]

(c)

Find the power at which the 25 N force is working.

-
.

[ 1 ]

[Maximum number: 4]

A car of mass 900 kg is moving along a straight horizontal road against a constant resistance to motion of 350 N . At an instant when the car is moving at $15 \mathrm{~ms}^{-1}$ its acceleration is $0.25 \mathrm{~ms}^{-2}$ .

(a)

Find the driving force of the car's engine at this instant.

[ 2 ]

(b)

Find the power of the car's engine at this instant.

[ 2 ]

[Maximum number: 5]

Figure for Question 1 — CAIE A-Level Mathematics

Coplanar forces of magnitudes $P \mathrm{~N}, Q \mathrm{~N}, 32 \mathrm{~N}$ and 21 N act at a point in the directions shown in the diagram. The forces are in equilibrium.

Find the value of P and the value of Q.

[Maximum number: 7]

An athlete runs along a straight horizontal road. The athlete starts from rest and accelerates at $0.5 \mathrm{~ms}^{-2}$ reaching a speed of $V \mathrm{~ms}^{-1}$ . The athlete maintains this speed of $V \mathrm{~ms}^{-1}$ for T s before decelerating at $0.2 \mathrm{~m} \mathrm{~s}^{-2}$ back to rest. The athlete covers a total distance of 1350 m in 246 seconds.

(a)

Sketch the velocity-time graph for the motion of the athlete.

Figure for Question 2(a) — CAIE A-Level Mathematics

[ 1 ]

(b)

Find T in terms of V, and hence show that 7 V^2-492 V+2700=0.

[ 4 ]

(c)

Find the value of V, giving a justification for your answer.

[ 2 ]

[Maximum number: 5]

Two particles, P and Q, of masses 3 kg and 5 kg respectively, are at rest on a smooth horizontal plane. P is projected at a speed of $4 \mathrm{~ms}^{-1}$ directly towards Q. After P and Q collide, P has speed $1 \mathrm{~ms}^{-1}$ .

(a)

Find the two possible speeds of Q after the collision.

[ 3 ]

(b)

It is given that $\lambda \mathrm{J}$ of kinetic energy, where $\lambda>0$ , is lost during the collision.

Find the value of $\lambda$ .

[ 2 ]

[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 ]

(c)

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

[ 1 ]

[Maximum number: 6]

Figure for Question 3 — CAIE A-Level Mathematics

Coplanar forces of magnitudes $45 \mathrm{~N}, 28 \mathrm{~N}, 72 \mathrm{~N}$ and 35 N act at a point in the directions shown in the diagram.

Find, in either order, the magnitude and direction of the resultant force.

[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 ]

(ii)

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

[ 2 ]

(iii)

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

[ 3 ]

(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 ]

[Maximum number: 7]

When a particle P of mass $m \mathrm{~kg}$ has speed $u \mathrm{~m} \mathrm{~s}^{-1}$ , its momentum is 4 Ns and its kinetic energy is 16 J.

(a)

Find the value of m and the value of u.

[ 3 ]

(b)

P is now projected on a smooth horizontal surface with speed v ms^-1 directly towards a particle Q of mass 1.25 kg which is stationary. After P and Q collide, the velocity of P is w ms^-1 and the velocity of Q is 2 w m s^-1. The loss of kinetic energy in the collision is 25 J.

Find the value of v and the value of w.

[ 4 ]

[Maximum number: 11]

A car of mass 1600 kg travels at constant speed $20 \mathrm{~m} \mathrm{~s}^{-1}$ up a straight road inclined at an angle of $\sin ^{-1} 0.12$ to the horizontal.

(a)

Find the change in potential energy of the car in 30 s .

[ 3 ]

(b)

Given that the total work done by the engine of the car in this time is 1960 kJ , find the constant force resisting the motion.

[ 3 ]

(c)

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

[ 2 ]

(d)

Given that this power is suddenly decreased by 15 %, find the instantaneous deceleration of the car.

[ 3 ]

Covered syllabus points

Question 1

Question 1(a)

Question 1(b)

Question 1(c)

Question 1

Question 1(a)

Question 1(b)

Question 1

Question 2

Question 2(a)

Question 2(b)

Question 2(c)

Question 2

Question 2(a)

Question 2(b)

Question 3

Question 3(a)

Question 3(b)

Question 3(c)

Question 3

Question 4

Question 4(a)

Question 4(a)(i)

Question 4(a)(ii)

Question 4(a)(iii)

Question 4(b)

Question 5

Question 5(a)

Question 5(b)

Question 5

Question 5(a)

Question 5(b)

Question 5(c)

Question 5(d)