Questions - Edexcel M3 - A-Level Maths

show that $u ^ { 2 } = 2 g \sin \theta$. The string $D P$ breaks when $P$ is at its lowest point. $P$ then immediately starts to move in a horizontal circle on the end of the string $C P$.
Prove that $\tan \theta = \frac { 1 } { 5 } \sqrt { 5 }$.

Edexcel M3 Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b85b908-bb74-4532-a1b4-3826946bd43b-2_341_652_217_621} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A particle of mass 0.6 kg is attached to one end of a light elastic spring of natural length 1 m and modulus of elasticity 30 N . The other end of the spring is fixed to a point $O$ which lies on a smooth plane inclined at an angle $\alpha$ to the horizontal where $\tan \alpha = \frac { 3 } { 4 }$ as shown in Figure 1. The particle is held at rest on the slope at a point 1.2 m from $O$ down the line of greatest slope of the plane.

Find the tension in the spring.
Find the initial acceleration of the particle.

Edexcel M3 Q2

2. A particle $P$ of mass 0.5 kg moves along the positive $x$-axis under the action of a single force directed away from the origin $O$. When $P$ is $x$ metres from $O$, the magnitude of the force is $3 x ^ { \frac { 1 } { 2 } } \mathrm {~N}$ and $P$ has a speed of $v \mathrm {~ms} ^ { - 1 }$. Given that when $x = 1 , P$ is moving away from $O$ with speed $2 \mathrm {~ms} ^ { - 1 }$,

find an expression for $v ^ { 2 }$ in terms of $x$,
show that when $x = 4 , P$ has a speed of $7.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to 1 decimal place.

Edexcel M3 Q3

3. A particle is performing simple harmonic motion along a straight line between the points $A$ and $B$ where $A B = 8 \mathrm {~m}$. The period of the motion is 12 seconds.

Find the maximum speed of the particle in terms of $\pi$. The points $P$ and $Q$ are on the line $A B$ at distances of 3 m and 6 m respectively from $A$.
Find, correct to 3 significant figures, the time it takes for the particle to travel directly from $P$ to $Q$.
(6 marks)

Edexcel M3 Q4

4. Whilst in free-fall a parachutist falls vertically such that his velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when he is $x$ metres below his initial position is given by $$v ^ { 2 } = k g \left( 1 - \mathrm { e } ^ { - \frac { 2 x } { k } } \right) ,$$ where $k$ is a constant.
Given that he experiences an acceleration of $\mathrm { f } \mathrm { m } \mathrm { s } ^ { - 2 }$,

show that $f = g \mathrm { e } ^ { - \frac { 2 x } { k } }$. After falling a large distance, his velocity is constant at $49 \mathrm {~ms} ^ { - 1 }$.
Find the value of $k$.
Hence, express $f$ in the form ( $\lambda - \mu v ^ { 2 }$ ) where $\lambda$ and $\mu$ are constants which you should find.
(4 marks)

Edexcel M3 Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b85b908-bb74-4532-a1b4-3826946bd43b-3_588_291_1126_662} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} A firework is modelled as a uniform solid formed by joining the plane surface of a right circular cone of height $2 r$ and base radius $r$, to one of the plane surfaces of a cylinder of height $h$ and base radius $r$ as shown in Figure 2. Using this model,

show that the distance of the centre of mass of the firework from its plane base is $$\frac { 3 h ^ { 2 } + 4 h r + 2 r ^ { 2 } } { 2 ( 3 h + 2 r ) }$$ The firework is to be launched from rough ground inclined at an angle $\alpha$ to the horizontal. Given that the firework does not slip or topple and that $h = 4 r$,
Find, correct to the nearest degree, the maximum value of $\alpha$.

Edexcel M3 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8b85b908-bb74-4532-a1b4-3826946bd43b-4_437_364_196_717} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} The two ends of a light inextensible string of length $3 a$ are attached to fixed points $Q$ and $R$ which are a distance of $a \sqrt { } 3$ apart with $R$ vertically below $Q$. A particle $P$ of mass $m$ is attached to the string at a distance of $2 a$ from $Q$.
$P$ is given a horizontal speed, $u$, such that it moves in a horizontal circle with both sections of the string taut as shown in Figure 3.

Show that $\angle P R Q$ is a right angle.
Find $\angle P Q R$ in degrees.
Find, in terms of $a , g , m$ and $u$, the tension in the section of string
1. $P Q$,
2. $P R$.
Show that $u ^ { 2 } \geq \frac { g a } { \sqrt { 3 } }$.

Edexcel M3 Q7

7. A particle of mass 2 kg is attached to one end of a light elastic string of natural length 1 m and modulus of elasticity 50 N . The other end of the string is attached to a fixed point $O$ on a rough horizontal plane and the coefficient of friction between the particle and the plane is $\frac { 10 } { 49 }$. The particle is projected from $O$ along the plane with an initial speed of $5 \mathrm {~ms} ^ { - 1 }$.

Show that the greatest distance from $O$ which the particle reaches is 1.84 m .
Find, correct to 2 significant figures, the speed at which the particle returns to $O$.

Edexcel M3 Q1

A student is attempting to model the expansion of an airbag in a car following a collision.

The student considers the displacement from the steering column, $s$ metres, of a point $P$ on the airbag $t$ seconds after a collision and uses the formula $$s = \mathrm { e } ^ { 3 t } - 1 , \quad 0 \leq t \leq 0.1$$ Using this model,

find, correct to the nearest centimetre, the maximum displacement of $P$,
find the initial velocity of $P$,
find the acceleration of $P$ in terms of $t$.
Explain why this model is unlikely to be realistic.

Edexcel M3 Q2

2. A particle $P$ is attached to one end of a light elastic string of modulus of elasticity 80 N . The other end of the string is attached to a fixed point $A$. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ad523c3f-9109-45a8-8399-80a4c2edeff7-2_410_570_1210_735} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} When a horizontal force of magnitude 20 N is applied to $P$, it rests in equilibrium with the string making an angle of $30 ^ { \circ }$ with the vertical and $A P = 1.2 \mathrm {~m}$ as shown in Figure 1.

Find the tension in the string.
Find the elastic potential energy stored in the string.

Edexcel M3 Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ad523c3f-9109-45a8-8399-80a4c2edeff7-3_513_570_196_625} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} A particle of mass $m$ is suspended at a point $A$ vertically below a fixed point $O$ by a light inextensible string of length $a$ as shown in Figure 2. The particle is given a horizontal velocity $u$ and subsequently moves along a circular arc until it reaches the point $B$ where the string becomes slack. Given that the point $B$ is at a height $\frac { 1 } { 2 } a$ above the level of $O$,

show that $\angle B O A = 120 ^ { \circ }$,
show that $u ^ { 2 } = \frac { 7 } { 2 } g a$.

Edexcel M3 Q4

4. On a particular day, high tide at the entrance to a harbour occurs at 11 a.m. and the water depth is 14 m . Low tide occurs $6 \frac { 1 } { 4 }$ hours later at which time the water depth is 6 m . In a model of the situation, the water level is assumed to perform simple harmonic motion.
Using this model,

write down the amplitude and period of the motion. A ship needs a depth of 9 m before it can enter or leave the harbour.
Show that on this day a ship must enter the harbour by 2.38 p.m., correct to the nearest minute, or wait for low tide to pass.
(6 marks)
Given that a ship is not ready to enter the harbour until 5 p.m.,
find, to the nearest minute, how long the ship must wait before it can enter the harbour.

Edexcel M3 Q5

5. (a) Use integration to show that the centre of mass of a uniform solid right circular cone of height $h$ is $\frac { 3 } { 4 } h$ from the vertex of the cone.
(6 marks) \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ad523c3f-9109-45a8-8399-80a4c2edeff7-4_419_424_372_721} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} A paperweight is made by removing material from the top half of a solid sphere of radius $r$ so that the remaining solid consists of a hemisphere of radius $r$ and a cone of height $r$ and base radius $r$ as shown in Figure 3.
(b) Find the distance of the centre of mass of the paperweight from its vertex.
(7 marks)

Edexcel M3 Q6

6. A car is travelling on a horizontal racetrack round a circular bend of radius 40 m . The coefficient of friction between the car and the road is $\frac { 2 } { 5 }$.

Find the maximum speed at which the car can travel round the bend without slipping, giving your answer correct to 3 significant figures.
(5 marks)
The owner of the track decides to bank the corner at an angle of $25 ^ { \circ }$ in order to enable the cars to travel more quickly.
Show that this increases the maximum speed at which the car can travel round the bend without slipping by 63\%, correct to the nearest whole number.
(8 marks)

Edexcel M3 Q7

7. A particle is travelling along the $x$-axis. At time $t = 0$, the particle is at $O$ and it travels such that its velocity, $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, at a distance $x$ metres from $O$ is given by $$v = \frac { 2 } { x + 1 }$$ The acceleration of the particle is $a \mathrm {~ms} ^ { - 2 }$.

Show that $a = \frac { - 4 } { ( x + 1 ) ^ { 3 } }$.
(4 marks) The points $A$ and $B$ lie on the $x$-axis. Given that the particle travels $d$ metres from $O$ to $A$ in $T$ seconds and 4 metres from $A$ to $B$ in 9 seconds,
show that $d = 1.5$,
find $T$.

Edexcel M3 Q1

A light elastic string has natural length $a$ and modulus of elasticity 4 mg . One end of the string is attached to a fixed point $A$ and a particle of mass $m$ is attached to the other end.

The particle is released from rest at $A$ and falls vertically until it comes to rest instantaneously at the point $B$. Find the distance $A B$ in terms of $a$.
(7 marks)

Edexcel M3 Q2

2. A particle $P$ of mass 0.25 kg is moving on a horizontal plane. At time $t$ seconds the velocity, $\mathbf { v } \mathrm { ms } ^ { - 1 }$, of $P$ relative to a fixed origin $O$ is given by $$\mathbf { v } = \ln ( t + 1 ) \mathbf { i } - \mathrm { e } ^ { - 2 t } \mathbf { j } , t \leq 0 ,$$ where $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors in the horizontal plane.

Find the acceleration of $P$ in terms of $t$.
Find, correct to 3 significant figures, the magnitude of the resultant force acting on $P$ when $t = 1$.
(4 marks)

Edexcel M3 Q3

3. A coin of mass 5 grams is placed on a vinyl disc rotating on a record player. The distance between the centre of the coin and the centre of the disc is 0.1 m and the coefficient of friction between the coin and the disc is $\mu$. The disc rotates at 45 revolutions per minute around a vertical axis at its centre and the coin moves with it and does not slide. By modelling the coin as a particle and giving your answers correct to an appropriate degree of accuracy, find

the speed of the coin,
the horizontal and vertical components of the force exerted on the coin by the disc. Given that the coin is on the point of moving,
show that, correct to 2 significant figures, $\mu = 0.23$.

Edexcel M3 Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cab238c9-f4e2-4637-a079-f74779548f49-3_447_506_205_657} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} A stand used to reach high shelves in a storeroom is in the shape of a frustum of a cone. It is modelled as a uniform solid formed by removing a right circular cone of height $2 h$ from a similar cone of height $3 h$ and base radius $3 r$ as shown in Figure 1.

Show that the centre of mass of the stand is a distance of $\frac { 33 } { 76 } h$ from its larger plane face.
(7 marks)
The stand is stored hanging in equilibrium from a point on the circumference of the larger plane face. Given that $h = 2 r$,
find, correct to the nearest degree, the acute angle which the plane faces of the stand make with the vertical.
(4 marks)

Edexcel M3 Q5

5. A particle of mass 0.8 kg is moving along the positive $x$-axis at a speed of $5 \mathrm {~ms} ^ { - 1 }$ away from the origin $O$. When the particle is 2 metres from $O$ it becomes subject to a single force directed towards $O$. The magnitude of the force is $\frac { k } { x ^ { 2 } } \mathrm {~N}$ when the particle is $x$ metres from $O$. Given that when the particle is 4 m from $O$ its speed has been reduced to $3 \mathrm {~ms} ^ { - 1 }$,

show that $k = \frac { 128 } { 5 }$,
find the distance of the particle from $O$ when it comes to instantaneous rest. (4 marks)

Edexcel M3 Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cab238c9-f4e2-4637-a079-f74779548f49-4_206_977_201_470} \captionsetup{labelformat=empty} \caption{Fig. 2}

\end{figure} Figure 2 shows a particle $P$ of mass $m$ which lies on a smooth horizontal table. It is attached to a point $A$ on the table by a light elastic spring of natural length $3 a$ and modulus of elasticity $\lambda$, and to a point $B$ on the table by a light elastic spring of natural length $2 a$ and modulus of elasticity $2 \lambda$. The distance between the points $A$ and $B$ is $7 a$.

Show that in equilibrium $A P = \frac { 9 } { 2 } a$. The particle is released from rest at a point $Q$ where $Q$ lies on the line $A B$ and $A Q = 5 a$.
Prove that the subsequent motion of the particle is simple harmonic with a period of $\pi \sqrt { \frac { 3 m a } { \lambda } }$.
(9 marks)

Edexcel M3 Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cab238c9-f4e2-4637-a079-f74779548f49-4_300_952_1201_497} \captionsetup{labelformat=empty} \caption{Fig. 3}

\end{figure} Figure 3 shows a vertical cross-section through part of a ski slope consisting of a horizontal section $A B$ followed by a downhill section $B C$. The point $O$ is on the same horizontal level as $C$ and $B C$ is a circular arc of radius 30 m and centre $O$, such that $\angle B O C = 90 ^ { \circ }$. A skier of mass 60 kg is skiing at $12 \mathrm {~ms} ^ { - 1 }$ along $A B$.

Assuming that friction and air resistance may be neglected, find the magnitude of the loss in reaction between the skier and the surface at $B$.
(4 marks)
The skier subsequently leaves the slope at the point $P$.
Find, correct to 3 significant figures, the speed at which the skier leaves the slope.
Find, correct to 3 significant figures, the speed of the skier immediately before hitting the ground again at the point $D$ which is on the same horizontal level as $C$.

Edexcel M3 Q1

The mechanism for releasing the ball on a pinball machine contains a light elastic spring of natural length 15 cm and modulus of elasticity $\lambda$.

The spring is held compressed to a length of 9 cm by a force of 4.5 N .

Find $\lambda$.
Find the work done in compressing the spring from a length of 9 cm to a length of 5 cm .
(4 marks)

Edexcel M3 Q2

2. A small bead $P$ is threaded onto a smooth circular wire of radius 0.8 m and centre $O$ which is fixed in a vertical plane. The bead is projected from the point vertically below $O$ with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and moves in complete circles about $O$.

Suggest a suitable model for the bead.
Given that the minimum speed of $P$ is $60 \%$ of its maximum speed, use the principle of conservation of energy to show that $u = 7$.
(6 marks)

Edexcel M3 Q3

3. At time $t$ seconds the acceleration, $a \mathrm {~ms} ^ { - 2 }$, of a particle is given by $$a = \frac { 4 } { ( 1 + t ) ^ { 3 } }$$ When $t = 0$, the particle has velocity $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and displacement 3 m from a fixed origin $O$.

Find an expression for the velocity of the particle in terms of $t$.
Show that when $t = 3$ the particle is 10.5 m from $O$.

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