Questions - WJEC - A-Level Maths

Show that $\theta = 30 ^ { \circ }$.
Calculate the length of each chain.

WJEC Further Unit 3 2019 June Q6

6. The diagram shows a rollercoaster at an amusement park where a car is projected from a launch point $O$ so that it performs a loop before instantaneously coming to rest at point $C$. The car then performs the same journey in reverse.
\includegraphics[max width=\textwidth, alt={}, center]{b430aa50-27e3-46f7-afef-7b8e75d46e1f-5_677_1733_552_166} The loop section is modelled by considering the track to be a vertical circle of radius 10 m and the car as a particle of mass $m$ kg moving on the inside surface of the circular loop. You may assume that the track is smooth. At point $A$, which is the lowest point of the circle, the car has velocity $u \mathrm {~ms} ^ { - 1 }$ such that $u ^ { 2 } = 60 g$. When the car is at point $B$ the radius makes an angle $\theta$ with the downward vertical.

Find, in terms of $\theta$ and $g$, an expression for $v ^ { 2 }$, where $v \mathrm {~ms} ^ { - 1 }$ is the speed of the car at $B$.
Show that $R \mathrm {~N}$, the reaction of the track on the car at $B$, is given by $$R = m g ( 4 + 3 \cos \theta ) .$$
Explain why the expression for $R$ in part (b) shows that the car will perform a complete loop.
This model predicts that the car will stop at $C$ at a vertical height of 30 m above $A$. However, after the car has completed the loop, the track becomes rough and the car only reaches a point $D$ at a vertical height of 28 m above $A$. The resistance to motion of the car beyond the loop is of constant magnitude $\frac { m g } { 32 } \mathrm {~N}$. Calculate the length of the rough track between $A$ and $D$.

WJEC Further Unit 3 2019 June Q7

7. Three spheres $A , B , C$, of equal radii and each of mass $m \mathrm {~kg}$, lie at rest on a smooth horizontal surface such that their centres are in a straight line with $B$ between $A$ and $C$. The coefficient of restitution between $A$ and $B$ is $e$. Sphere $A$ is projected towards $B$ with speed $u \mathrm {~ms} ^ { - 1 }$ so that it collides with $B$.

Find expressions, in terms of $e$ and $u$, for the speed of $A$ and the speed of $B$ after they collide. You are now given that $e = \frac { 1 } { 2 }$.
Find, in terms of $m$ and $u$, the loss in kinetic energy due to the collision between $A$ and $B$.
After the collision between $A$ and $B$, sphere $B$ then collides with $C$. The coefficient of restitution between $B$ and $C$ is $e _ { 1 }$. Show that there will be no further collisions if $e _ { 1 } \leqslant \frac { 1 } { 3 }$.

WJEC Further Unit 3 2022 June Q1

A particle of mass 1.2 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is fixed to a point $O$ on a smooth horizontal surface. With the string taut, the particle moves on the surface with constant speed $8 \mathrm {~ms} ^ { - 1 }$ in a horizontal circle with centre $O$.
1. Find the angular velocity of the particle about $O$.
2. Calculate the tension in the string.
3. The diagram below shows a woman standing at the end of a diving platform. She is about to dive into the water below.
The woman has mass 60 kg and she may be modelled as a particle positioned at the end of the platform which is 10 m above the water.
\includegraphics[max width=\textwidth, alt={}, center]{2954c34b-dd8f-49a2-a379-704022574550-2_750_942_1338_566} When the woman dives, she projects herself from the platform with a speed of $7.8 \mathrm {~ms} ^ { - 1 }$.
Find the kinetic energy of the woman when she leaves the platform.
Initially, the situation is modelled ignoring air resistance. By using conservation of energy, show that the model predicts that the woman enters the water with an approximate speed of $16 \mathrm {~ms} ^ { - 1 }$.
Suppose that this model is refined to include air resistance so that the speed with which the woman enters the water is now predicted to be $13 \mathrm {~ms} ^ { - 1 }$. Determine the amount of energy lost to air resistance according to the refined model.

WJEC Further Unit 3 2022 June Q3

3. Two spheres $A$ and $B$, of equal radii, are moving towards each other on a smooth horizontal surface and collide directly. Sphere $A$ has mass $4 m \mathrm {~kg}$ and sphere $B$ has mass $3 m \mathrm {~kg}$. Just before the collision, $A$ has speed $9 \mathrm {~ms} ^ { - 1 }$ and $B$ has speed $3.5 \mathrm {~ms} ^ { - 1 }$. Immediately after the collision, $A$ has speed $1.5 \mathrm {~ms} ^ { - 1 }$ in the direction of its original motion.

Show that the speed of $B$ immediately after the collision is $6.5 \mathrm {~ms} ^ { - 1 }$.
Calculate the coefficient of restitution between $A$ and $B$.
Given that the magnitude of the impulse exerted by $B$ on $A$ is 36 Ns , find the value of $m$.
Give a reason why it is not necessary to model the spheres as particles in this question.

WJEC Further Unit 3 2022 June Q4

4. A particle $P$ of mass 0.5 kg is in equilibrium under the action of three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$. $$\mathbf { F } _ { 1 } = ( 9 \mathbf { i } + 6 \mathbf { j } - 12 \mathbf { k } ) \mathbf { N } \quad \text { and } \quad \mathbf { F } _ { 2 } = ( 6 \mathbf { i } - 7 \mathbf { j } + 3 \mathbf { k } ) \mathbf { N } .$$

Find the force $\mathbf { F } _ { 3 }$.
Forces $\mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ are removed so that $P$ moves in a straight line $A B$ under the action of the single force $\mathbf { F } _ { 1 }$. The points $A$ and $B$ have position vectors ( $2 \mathbf { i } - 9 \mathbf { j } + 7 \mathbf { k }$ )m and $( 8 \mathbf { i } - 5 \mathbf { j } - \mathbf { k } ) \mathrm { m }$ respectively. The particle $P$ is initially at rest at $A$.
1. Verify that $\mathbf { F } _ { 1 }$ acts parallel to the vector $\mathbf { A B }$.
2. Find the work done by the force $\mathbf { F } _ { 1 }$ as the particle moves from $A$ to $B$.
3. By using the work-energy principle, find the speed of $P$ as it reaches $B$.

WJEC Further Unit 3 2022 June Q5

5. One end of a light elastic string, of natural length 2.5 m and modulus of elasticity $30 g \mathrm {~N}$, is fixed to a point $O$. A particle $P$, of mass 2 kg , is attached to the other end of the string. Initially, $P$ is held at rest at the point $O$. It is then released and allowed to fall under gravity.

Show that, while the string is taut, $$v ^ { 2 } = g \left( 5 + 2 x - 6 x ^ { 2 } \right)$$ where $v \mathrm {~ms} ^ { - 1 }$ denotes the velocity of the particle when the extension in the string is $x \mathrm {~m}$.
Calculate the maximum extension of the string.
1. Find the extension of the string when $P$ attains its maximum speed.
2. Hence determine the maximum speed of $P$.

WJEC Further Unit 3 2022 June Q6

6. A vehicle of mass 3500 kg is moving up a slope inclined at an angle $\alpha$ to the horizontal. When the vehicle is travelling at a velocity of $v \mathrm {~ms} ^ { - 1 }$, the resistance to motion can be modelled by a variable force of magnitude $40 \nu \mathrm {~N}$.

Given that $\sin \alpha = \frac { 3 } { 49 }$, calculate the power developed by the engine at the instant when the speed of the vehicle is $25 \mathrm {~ms} ^ { - 1 }$ and its deceleration is $0.2 \mathrm {~ms} ^ { - 2 }$.
When the vehicle's engine is working at a constant rate of 40 kW , the maximum speed that can be maintained up the slope is $20 \mathrm {~ms} ^ { - 1 }$. Find the value of $\alpha$. Give your answer in degrees, correct to one significant figure.

WJEC Further Unit 3 2022 June Q7

7. The diagram below shows a particle $P$, of mass $2 \cdot 5 \mathrm {~kg}$, attached by means of two light inextensible strings fixed at points $A$ and $B$. Point $A$ is vertically above point $B$. $B P$ makes an angle of $60 ^ { \circ }$ with the upward vertical and $A P$ is inclined at an angle $\theta$ to the downward vertical where $\cos \theta = 0 \cdot 8$. The particle $P$ describes a horizontal circle with constant angular speed $\omega$ radians per second about centre $C$ with both strings taut.
\includegraphics[max width=\textwidth, alt={}, center]{2954c34b-dd8f-49a2-a379-704022574550-5_823_641_644_705} The tension in the string $B P$ is $39 \cdot 2 \mathrm {~N}$.

Calculate the tension in the string $A P$.
Given that the length of the string $A P$ is 1.5 m , find the value of $\omega$.
Calculate the kinetic energy of $P$.

WJEC Further Unit 3 2023 June Q1

One end of a light elastic string, of natural length $0 \cdot 2 \mathrm {~m}$ and modulus of elasticity $5 g \mathrm {~N}$, is attached to a fixed point $O$. The other end is attached to a particle of mass 4 kg . The particle hangs in equilibrium vertically below $O$.
1. Show that the extension of the string is 0.16 m .
2. The particle is pulled down vertically and held at rest so that the extension of the string is 0.28 m . The particle is then released. Determine the speed of the particle as it passes through the equilibrium position.
3. At time $t = 0$ seconds, a particle $A$ has position vector $( 6 \mathbf { i } + 21 \mathbf { j } - 8 \mathbf { k } )$ m relative to a fixed origin O and is moving with constant velocity $( 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } ) \mathrm { ms } ^ { - 1 }$.
4. Write down the position vector of particle $A$ at time $t$ seconds and hence find the distance $O A$ when $t = 5$.
5. The position vector, $\mathbf { r } _ { B }$ metres, of another particle $B$ at time $t$ seconds is given by
$$\mathbf { r } _ { B } = 3 \sin \left( \frac { t } { 2 } \right) \mathbf { i } - 3 \cos \left( \frac { t } { 2 } \right) \mathbf { j } + 5 \mathbf { k } .$$ (i) Show that $B$ is moving with constant speed.
(ii) Determine the smallest value of $t$ such that particles $A$ and $B$ are moving perpendicular to each other.

WJEC Further Unit 3 2023 June Q3

3. The diagram below shows a hollow cone, of base radius 5 m and height 12 m , which is fixed with its axis vertical and vertex $V$ downwards. A particle $P$, of mass $M \mathrm {~kg}$, moves in a horizontal circle with centre $C$ on the smooth inner surface of the cone with constant speed $v = 3 \sqrt { g } \mathrm {~ms} ^ { - 1 }$.
\includegraphics[max width=\textwidth, alt={}, center]{b6b77d92-acb6-4cc3-8328-1899ed4a87cd-4_940_695_598_683}

Show that the normal reaction of the surface of the cone on the particle is $\frac { 13 M g } { 5 } \mathrm {~N}$.
Calculate the length of $C P$ and hence determine the height of $C$ above $V$.

WJEC Further Unit 3 2023 June Q4

4. Geraint is a cyclist competing in a race along the Taff Trail. The Taff Trail is a track that runs from Cardiff Bay to Brecon. The chart below shows the altitude (height above sea level) along the route. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Altitude Chart of the Taff Trail} \includegraphics[alt={},max width=\textwidth]{b6b77d92-acb6-4cc3-8328-1899ed4a87cd-5_620_1701_632_191}

\end{figure} Geraint starts from rest at Cardiff Bay and has a speed of $10 \mathrm {~ms} ^ { - 1 }$ when he crosses the finish line in Brecon. Geraint and his bike have a total mass of 80 kg . The resistance to motion may be modelled by a constant force of magnitude 16 N .

Given that 1440 kJ of energy is used in overcoming resistances during the race,
1. find the length of the track,
2. calculate the work done by Geraint.
The steepest section of the track may be modelled as a slope inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 2 } { 7 }$.
\includegraphics[max width=\textwidth, alt={}, center]{b6b77d92-acb6-4cc3-8328-1899ed4a87cd-5_513_844_1987_607} Geraint is capable of producing a maximum power of 250 W . Find the maximum speed that Geraint can attain whilst travelling on this section of the track.

WJEC Further Unit 3 2023 June Q5

5. The diagram below shows two spheres $A$ and $B$, of equal radii, moving in the same direction on a smooth horizontal surface. Sphere $A$, of mass 3 kg , is moving with speed $4 \mathrm {~ms} ^ { - 1 }$ and sphere $B$, of mass 2 kg , is moving with speed $10 \mathrm {~ms} ^ { - 1 }$.
\includegraphics[max width=\textwidth, alt={}, center]{b6b77d92-acb6-4cc3-8328-1899ed4a87cd-6_275_972_580_548} Sphere $B$ is then given an impulse after which it moves in the opposite direction with speed $6 \mathrm {~ms} ^ { - 1 }$.

Calculate the magnitude and direction of the impulse exerted on $B$. Sphere $B$ continues to move with speed $6 \mathrm {~ms} ^ { - 1 }$ so that it collides directly with sphere $A$. The kinetic energy lost due to the collision is 45 J .
Calculate the speed of $A$ and the speed of $B$ immediately after the two spheres collide. State the direction in which each sphere is moving relative to its motion immediately before the collision.

WJEC Further Unit 3 2023 June Q6

6. The diagram shows a slide, $A B C$, at a water park. The shape of the slide may be modelled by two circular arcs, $A B$ and $B C$, in the same vertical plane. Arc $A B$ has radius 7 m and subtends an angle $\alpha$ at its centre $D$, where $\cos \alpha = \frac { 9 } { 10 }$. Arc $B C$ has radius 5 m and subtends an angle of $45 ^ { \circ }$ at its centre, $O$. The straight line $D B O$ is vertical.
\includegraphics[max width=\textwidth, alt={}, center]{b6b77d92-acb6-4cc3-8328-1899ed4a87cd-7_1171_940_630_552} Users of the slide are required to sit in a rubber ring and are released from rest at point $A$. A girl decides to use the slide. The combined mass of the girl and the rubber ring is 50 kg .

When the rubber ring is at a point $P$ on the circular arc $B C$, its speed is $v \mathrm {~ms} ^ { - 1 }$ and $O P$ makes an angle $\theta$ with the upward vertical.
1. Show that $v ^ { 2 } = 111.72 - 98 \cos \theta$.
2. Find, in terms of $\theta$, the reaction between the rubber ring and the slide at $P$.
3. Show that, according to this model, the rubber ring loses contact with the slide before reaching $C$.
4. In reality, there will be resistive forces opposing the motion of the rubber ring. Explain how this fact will affect your answer to (iii).
Show that the rubber ring will remain in contact with the slide along the arc $A B$.

WJEC Further Unit 3 2024 June Q1

4 marks

1. Two particles $A$ and $B$, of masses 2 kg and 5 kg respectively, are moving in the same direction along a smooth horizontal surface when they collide directly. Before the collision, $B$ is moving with speed $1.2 \mathrm {~ms} ^ { - 1 }$ and, immediately after the collision, its speed is $3.8 \mathrm {~ms} ^ { - 1 }$. The coefficient of restitution between the particles $A$ and $B$ is 0.3 .

1. Find the impulse exerted by $A$ on $B$.
2. Given that the particles $A$ and $B$ were in contact for 0.08 seconds, find the average force between $A$ and $B$.
  [0pt] [4]
  \end{tabular} & Examiner only
  \hline \end{tabular} \end{center}
Calculate the speed of $A$ before and after the collision.
After the collision between $A$ and $B$, particle $B$ continues to move with speed $3.8 \mathrm {~ms} ^ { - 1 }$ until it collides directly with a stationary particle $C$ of mass 4 kg . When $B$ and $C$ collide, they coalesce to form a single particle.
1. Write down the coefficient of restitution between $B$ and $C$.
2. Determine the speed of the combined particle after the collision.
  \section*{PLEASE DO NOT WRITE ON THIS PAGE}

WJEC Further Unit 3 2024 June Q2

The diagram below shows a light spring of natural length 1.2 m and modulus of elasticity 84 N . One end of the spring $A$ is fixed and the other end is attached to an object $P$ of mass 4 kg .
\includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-06_542_451_466_808}

Initially, $P$ is held at rest with the spring stretched to a total length of 2.2 m and $A P$ vertical.

Show that the elastic energy stored in the spring is 35 J .
The object $P$ is then released. Find the speed of $P$ at the instant when the elastic energy in the spring is reduced to $5 \cdot 6 \mathrm {~J}$.

WJEC Further Unit 3 2024 June Q3

3. Three forces $( 4 \mathbf { i } - 7 \mathbf { j } + 9 \mathbf { k } ) \mathrm { N } , ( 5 \mathbf { i } + 3 \mathbf { j } - 8 \mathbf { k } ) \mathrm { N }$ and $( - 2 \mathbf { i } + 6 \mathbf { j } - 11 \mathbf { k } ) \mathrm { N }$ act on a particle.

Find the resultant $\mathbf { R }$ of the three forces.
The points $A$ and $B$ have position vectors $( 3 \mathbf { i } + 4 \mathbf { j } - 12 \mathbf { k } ) \mathrm { m }$ and $( a \mathbf { i } + 7 \mathbf { j } - 10 \mathbf { k } ) \mathrm { m }$ respectively, where $a$ is a constant. The work done by $\mathbf { R }$ in moving the particle from $A$ to $B$ is 21 J . Calculate the value of $a$.
\section*{PLEASE DO NOT WRITE ON THIS PAGE}

WJEC Further Unit 3 2024 June Q4

The diagram below shows a particle $P$, of mass 5 kg , attached to one end of a light inextensible string of length 3 m . The other end is fixed at a point $A$. The particle $P$ is moving in a horizonal circle with centre $C$, where the point $C$ is vertically below $A$. The string is inclined at an angle $\theta$ to the downward vertical, where $\tan \theta = \frac { 20 } { 21 }$.
\includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-10_725_796_639_628}

Find the speed of the particle.

WJEC Further Unit 3 2024 June Q5

5. A particle of mass 2 kg is moving under the action of a force $\mathbf { F N }$ which, at time $t$ seconds, is given by $$\mathbf { F } = 4 t \mathbf { i } - \sqrt { t } \mathbf { j } + 6 \mathbf { k }$$ When $t = 1$, the velocity of the particle is $\left( 3 \mathbf { i } - \frac { 1 } { 3 } \mathbf { j } - \mathbf { k } \right) \mathrm { ms } ^ { - 1 }$.

Find an expression for the velocity vector of the particle at time $t \mathrm {~s}$.
Determine the values of $t$ when the particle is moving in a direction perpendicular to the vector $( - \mathbf { i } + 3 \mathbf { k } )$.

WJEC Further Unit 3 2024 June Q6

6. A slope is inclined at an angle of $5 ^ { \circ }$ to the horizontal. A car, of mass 1500 kg , has an engine that is working at a constant rate of $P \mathrm {~W}$. The resistance to motion of the car is constant at 4500 N . When the car is moving up the slope, its acceleration is $a \mathrm {~ms} ^ { - 2 }$ at the instant when its speed is $10 \mathrm {~ms} ^ { - 1 }$. When the car is moving down the slope, its deceleration is $a \mathrm {~ms} ^ { - 2 }$ at the instant when its speed is $20 \mathrm {~ms} ^ { - 1 }$. Determine the value of $P$ and the value of $a$.
\includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-14_87_1609_635_267}

WJEC Further Unit 3 2024 June Q7

7. One end of a light rod of length $\frac { 5 } { 7 } \mathrm {~m}$ is attached to a fixed point $O$ and the other end is attached to a particle $P$, of mass $m \mathrm {~kg}$. The particle $P$ is projected from the point $A$, which is vertically below $O$, with a horizontal speed of $u \mathrm {~ms} ^ { - 1 }$ so that it moves in a vertical circle with centre $O$. When the rod $O P$ is inclined at an angle $\theta$ to the downward vertical, the speed of $P$ is $v \mathrm {~ms} ^ { - 1 }$ and the tension in the rod is $T \mathrm {~N}$.
\includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-16_629_593_646_735}

Show that $$v ^ { 2 } = u ^ { 2 } - 14 + 14 \cos \theta$$
Hence determine the least possible value of $u ^ { 2 }$ for the particle to reach the highest point of the circle.
Given that $u ^ { 2 } = 32 \cdot 2$,
1. find, in terms of $m$ and $\theta$, an expression for $T$,
2. calculate the range of values of $\theta$ such that the rod is exerting a thrust.
  State whether your answer to (c)(ii) would be different if the mass of the particle was reduced. Give a reason for your answer. Additional page, if required. Write the question number(s) in the left-hand margin. only

WJEC Further Unit 3 Specimen Q1

\begin{enumerate} \item By burning a charge, a cannon fires a cannon ball of mass 12 kg horizontally. As the cannon ball leaves the cannon, its speed is $600 \mathrm {~ms} ^ { - 1 }$. The recoiling part of the cannon has a mass of 1600 kg .

Determine the speed of the recoiling part immediately after the cannon ball leaves the cannon.
Find the energy created by the burning of the charge. State any assumption you have made in your solution and briefly explain how the assumption affects your answer.
Calculate the constant force needed to bring the recoiling part to rest in 1.2 m . State, with a reason, whether your answer is an overestimate or an underestimate of the actual force required. \item A particle $P$, of mass 3 kg , is attached to a fixed point $O$ by a light inextensible string of length 4 m . Initially, particle $P$ is held at rest at a point which is $2 \sqrt { 3 } \mathrm {~m}$ horizontally from $O$. It is then released and allowed to fall under gravity.

WJEC Further Unit 3 Specimen Q5

5. A particle of mass $m \mathrm {~kg}$ is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point $O$. The particle is set in motion such that it moves in a horizontal circle of radius 2 m with constant speed $4.8 \mathrm {~ms} ^ { - 1 }$. Calculate the angle the string makes with the vertical.

WJEC Further Unit 3 Specimen Q6

6. A particle of mass 5 kg is attached to a string $A B$ and a rod $B C$ at the point $B$. The string $A B$ is light and elastic with modulus $\lambda \mathrm { N }$ and natural length 2 m . The rod $B C$ is light and of length 2 m . The end $A$ of the string is attached to a fixed point and the end $C$ of the rod is attached to another fixed point such that $A$ is vertically above $C$ with $A C = 2 \mathrm {~m}$. When the particle rests in equilibrium, $A B$ makes an angle of $50 ^ { \circ }$ with the downward vertical.

Determine, in terms of $\lambda$, the tension in the string $A B$.
Calculate, in terms of $\lambda$, the energy stored in the string $A B$.
Find, in terms of $\lambda$, the thrust in the rod $B C$.

WJEC Further Unit 3 Specimen Q7

7. A vehicle of mass 6000 kg is moving up a slope inclined at an angle $\alpha$ to the horizontal, where $\sin \alpha = \frac { 6 } { 49 }$. The vehicle's engine exerts a constant power of $P \mathrm {~W}$. The constant resistance to motion of the vehicle is $R \mathrm {~N}$. At the instant the vehicle is moving with velocity $\frac { 16 } { 5 } \mathrm {~ms} ^ { - 1 }$, its acceleration is $2 \mathrm {~ms} ^ { - 2 }$. The maximum velocity of the vehicle is $\frac { 16 } { 3 } \mathrm {~ms} ^ { - 1 }$. Determine the value of $P$ and the value of $R$.

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