Questions - SPS SPS SM Mechanics

$\alpha$,
$F$.

SPS SPS SM Mechanics 2021 May Q2

2. A particle $P$ moves with constant acceleration $( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }$. At time $t = 0$ seconds $P$ is at the origin. At time $t = 4$ seconds $P$ has velocity $( 2 \mathbf { i } + 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$.

Find the displacement vector of $P$ at time $t = 4$ seconds.
Find the speed of $P$ at time $t = 0$ seconds.

SPS SPS SM Mechanics 2021 May Q3

3. A uniform plank $A B$ has weight 100 N and length 4 m . The plank rests horizontally in equilibrium on two smooth supports $C$ and $D$, where $A C = x \mathrm {~m}$ and $C D = 0.5 \mathrm {~m}$ (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{14759a7d-dbec-4d72-a722-c83043fb59c5-08_184_1266_283_402} The magnitude of the reaction of the support on the plank at $C$ is 75 N . Modelling the plank as a rigid rod, find

the magnitude of the reaction of the support on the plank at $D$,
the value of $x$. A stone block, which is modelled as a particle, is now placed at the end of the plank at $B$ and the plank is on the point of tilting about $D$.
Find the weight of the stone block.
Explain the limitation of modelling
(a) the stone block as a particle,
(b) the plank as a rigid rod.

SPS SPS SM Mechanics 2021 May Q4

4. A particle $P$ projected from a point $O$ on horizontal ground hits the ground after 2.4 seconds. The horizontal component of the initial velocity of $P$ is $\frac { 5 } { 3 } d \mathrm {~ms} ^ { - 1 }$.

Find, in terms of $d$, the horizontal distance of $P$ from $O$ when it hits the ground.
Find the vertical component of the initial velocity of $P$.
$P$ just clears a vertical wall which is situated at a horizontal distance $d \mathrm {~m}$ from $O$.
Find the height of the wall. The speed of $P$ as it passes over the wall is $16 \mathrm {~ms} ^ { - 1 }$.
Find the value of $d$ correct to 3 significant figures.

SPS SPS SM Mechanics 2021 May Q5

5.
\includegraphics[max width=\textwidth, alt={}, center]{14759a7d-dbec-4d72-a722-c83043fb59c5-12_501_880_132_614} Two particles $P$ and $Q$ are attached to opposite ends of a light inextensible string which passes over a small smooth pulley at the top of a rough plane inclined at $30 ^ { \circ }$ to the horizontal. $P$ has mass 0.2 kg and is held at rest on the plane. $Q$ has mass 0.2 kg and hangs freely. The string is taut (see diagram). The coefficient of friction between $P$ and the plane is 0.4 . The particle $P$ is released.

State the tension in the string before $P$ is released, and find the tension in the string after $P$ is released.
$Q$ strikes the floor and remains at rest. $P$ continues to move up the plane for a further distance of 0.8 m before it comes to rest. $P$ does not reach the pulley.
Find the speed of the particles immediately before $Q$ strikes the floor.
Calculate the magnitude of the contact force exerted on $P$ by the plane while $P$ is in motion. End of Examination Extra Answer Space

SPS SPS SM Mechanics 2021 September Q1

A racing car starts from rest at the point $A$ and moves with constant acceleration.of $11 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ for 8 s . The velocity it has reached after 8 s is then maintained for $T \mathrm {~s}$. The racing car then decelerates from this velocity to $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in a further 2 s , reaching point $B$.
a Sketch a velocity-time graph to illustrate the motion of the racing car. Include the top speed of the racing car in your sketch.
b Given that the distance between $A$ and $B$ is 1404 m , find the value of $T$.
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A particle $P$ is acted upon by three forces $\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }$ and $\mathbf { F } _ { 3 }$ given by $\mathbf { F } _ { 1 } = ( 6 \mathbf { i } - 4 \mathbf { j } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( - 2 \mathbf { i } + 9 \mathbf { j } ) \mathrm { N }$ and $\mathbf { F } _ { 3 } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }$, where $a$ and $b$ are constants. Given that $P$ is in equilibrium,
a find the value of $a$ and the value of $b$.

The force $\mathbf { F } _ { 2 }$ is now removed. The resultant of $\mathbf { F } _ { 1 }$ and $\mathbf { F } _ { 3 }$ is $\mathbf { R }$.
b Find the magnitude of $\mathbf { R }$.
c Find the angle, to $0.1 ^ { \circ }$, that $\mathbf { R }$ makes with $\mathbf { i }$.
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SPS SPS SM Mechanics 2021 September Q3

3. A car of mass 1200 kg pulls a trailer of mass 400 kg along a straight horizontal road. The car and trailer are connected by a tow-rope modelled as a light inextensible rod. The engine of the car provides a constant driving force of 3200 N . The horizontal resistances of the car and the trailer are proportional to their respective masses. Given that the acceleration of the car and the trailer is $0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$,
a find the resistance to motion on the trailer,
b find the tension in the tow-rope. When the car and trailer are travelling at $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ the tow-rope breaks. Assuming that the resistances to motion remain unchanged,
c find the distance the trailer travels before coming to a stop,
d state how you have used the modelling assumption that the tow-rope is inextensible.
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SPS SPS SM Mechanics 2021 September Q4

4. A car starts from the point $A$. At time $t \mathrm {~s}$ after leaving $A$, the distance of the car from $A$ is $s \mathrm {~m}$, where $s = 30 t - 0.4 t ^ { 2 } , 0 \leqslant t \leqslant 25$. The car reaches the point $B$ when $t = 25$.
a Find the distance $A B$.
b Show that the car travels with a constant acceleration and state the value of this acceleration. A runner passes through $B$ when $t = 0$ with an initial velocity of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ running directly towards $A$. The runner has a constant acceleration of $0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
c Find the distance from $A$ at which the runner and the car pass one another.
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SPS SPS SM Mechanics 2022 February Q1

1. Answer all the questions.
Find $$\int \left( x ^ { 4 } - 6 x ^ { 2 } + 7 \right) \mathrm { d } x$$ giving your answer in simplest form.
(3)
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SPS SPS SM Mechanics 2022 February Q2

2. Given that $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 5 \quad x \in \mathbb { R }$$

express $\mathrm { f } ( x )$ in the form $( x + a ) ^ { 2 } + b$ where $a$ and $b$ are integers to be found. The curve with equation $y = \mathrm { f } ( x )$
- meets the $y$-axis at the point $P$
- has a minimum turning point at the point $Q$
- Write down
  1. the coordinates of $P$
  2. the coordinates of $Q$
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SPS SPS SM Mechanics 2022 February Q3

3. The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $$u _ { n + 1 } = k - \frac { 24 } { u _ { n } } \quad u _ { 1 } = 2$$ where $k$ is an integer.
Given that $u _ { 1 } + 2 u _ { 2 } + u _ { 3 } = 0$

show that $$3 k ^ { 2 } - 58 k + 240 = 0$$
Find the value of $k$, giving a reason for your answer.
Find the value of $u _ { 3 }$
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SPS SPS SM Mechanics 2022 February Q4

4. Relative to a fixed origin $O$,

the point $A$ has position vector $5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }$
the point $B$ has position vector $7 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$
the point $C$ has position vector $4 \mathbf { i } + 8 \mathbf { j } - 3 \mathbf { k }$
1. Find $| \overrightarrow { A B } |$ giving your answer as a simplified surd.

Given that $A B C D$ is a parallelogram,

find the position vector of the point $D$. The point $E$ is positioned such that

$A C E$ is a straight line
$A C : C E = 2 : 1$
Find the coordinates of the point $E$.
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\section*{Solutions relying entirely on calculator technology are not acceptable.} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{eacf7695-44c4-4937-8e92-5b0df8ad5f70-12_855_1104_340_589} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of part of the curve $C$ with equation $$y = x ^ { 3 } - 10 x ^ { 2 } + 27 x - 23$$ The point $P ( 5 , - 13 )$ lies on $C$
The line $l$ is the tangent to $C$ at $P$

Use differentiation to find the equation of $l$, giving your answer in the form $y = m x + c$ where $m$ and $c$ are integers to be found.

Hence verify that $l$ meets $C$ again on the $y$-axis. The finite region $R$, shown shaded in Figure 2, is bounded by the curve $C$ and the line $l$.

Use algebraic integration to find the exact area of $R$.
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SPS SPS SM Mechanics 2022 February Q6

6. A scientist is studying the growth of two different populations of bacteria. The number of bacteria, $N$, in the first population is modelled by the equation $$N = A \mathrm { e } ^ { k t } \quad t \geqslant 0$$ where $A$ and $k$ are positive constants and $t$ is the time in hours from the start of the study.
Given that

there were 1000 bacteria in this population at the start of the study
it took exactly 5 hours from the start of the study for this population to double
1. find a complete equation for the model.
2. Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures.

The number of bacteria, $M$, in the second population is modelled by the equation $$M = 500 \mathrm { e } ^ { 1.4 k t } \quad t \geqslant 0$$ where $k$ has the value found in part (a) and $t$ is the time in hours from the start of the study.
Given that $T$ hours after the start of the study, the number of bacteria in the two different populations was the same,

find the value of $T$.
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Show that $$\frac { 1 - \cos 2 \theta } { \sin ^ { 2 } 2 \theta } \equiv k \sec ^ { 2 } \theta \quad \theta \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$ where $k$ is a constant to be found.

Hence solve, for $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$ $$\frac { 1 - \cos 2 x } { \sin ^ { 2 } 2 x } = ( 1 + 2 \tan x ) ^ { 2 }$$ Give your answers to 3 significant figures where appropriate.
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SPS SPS SM Mechanics 2022 February Q8

8. Show that $$\sum _ { n = 2 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { n } \cos ( 180 n ) ^ { \circ } = \frac { 9 } { 28 }$$ [BLANK PAGE]

SPS SPS SM Mechanics 2022 February Q9

9. The function f is defined by $$\mathrm { f } ( x ) = \frac { ( x + 5 ) ( x + 1 ) } { ( x + 4 ) } - \ln ( x + 4 ) \quad x \in \mathbb { R } \quad x > k$$

State the smallest possible value of $k$.
Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { a x ^ { 2 } + b x + c } { ( x + 4 ) ^ { 2 } }$$ where $a$, $b$ and $c$ are integers to be found.
Hence show that f is an increasing function.
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SPS SPS SM Mechanics 2022 February Q10

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{eacf7695-44c4-4937-8e92-5b0df8ad5f70-24_707_716_255_826} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of the graph with equation $$y = | 2 x - 3 k |$$ where $k$ is a positive constant.

Sketch the graph with equation $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = k - | 2 x - 3 k |$$ stating
- the coordinates of the maximum point
- the coordinates of any points where the graph cuts the coordinate axes
- Find, in terms of $k$, the set of values of $x$ for which
$$k - | 2 x - 3 k | > x - k$$ giving your answer in set notation.
Find, in terms of $k$, the coordinates of the minimum point of the graph with equation $$y = 3 - 5 \mathrm { f } \left( \frac { 1 } { 2 } x \right)$$ [BLANK PAGE]
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SPS SPS SM Mechanics 2022 February Q11

11. The curve $C$ has parametric equations $$x = \sin 2 \theta \quad y = \operatorname { cosec } ^ { 3 } \theta \quad 0 < \theta < \frac { \pi } { 2 }$$

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$
Hence find the exact value of the gradient of the tangent to $C$ at the point where $y = 8$
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SPS SPS SM Mechanics 2022 February Q12

12. Answer all the questions.
Two cyclists, $A$ and $B$, are cycling along the same straight horizontal track.
The cyclists are modelled as particles and the motion of the cyclists is modelled as follows:

At time $t = 0$, cyclist $A$ passes through the point $O$ with speed $2 \mathrm {~ms} ^ { - 1 }$
Cyclist $A$ is moving in a straight line with constant acceleration $2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
At time $t = 2$ seconds, cyclist $B$ starts from rest at $O$
Cyclist $B$ moves with constant acceleration $6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ along the same straight line and in the same direction as cyclist $A$
At time $t = T$ seconds, $B$ overtakes $A$ at the point $X$

Using the model,

sketch, on the same axes, for the interval from $t = 0$ to $t = T$ seconds,
- a velocity-time graph for the motion of $A$
- a velocity-time graph for the motion of $B$
- explain why the two graphs must cross before time $t = T$ seconds,
- find the time when $A$ and $B$ are moving at the same speed,
- find the distance $O X$
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SPS SPS SM Mechanics 2022 February Q13

13.

\includegraphics[max width=\textwidth, alt={}]{eacf7695-44c4-4937-8e92-5b0df8ad5f70-34_328_1520_132_251}

A golfer hits a ball from a point $A$ with a speed of $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $15 ^ { \circ }$ above the horizontal. While the ball is in the air, it is modelled as a particle moving under the influence of gravity. Take the acceleration due to gravity to be $10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The ball first lands at a point $B$ which is 4 m below the level of $A$ (see diagram).

Determine the time taken for the ball to travel from $A$ to $B$.
Determine the horizontal distance of $B$ from $A$.
Determine the direction of motion of the ball 1.5 seconds after the golfer hits the ball.
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SPS SPS SM Mechanics 2022 February Q14

14.
\includegraphics[max width=\textwidth, alt={}, center]{eacf7695-44c4-4937-8e92-5b0df8ad5f70-36_527_1123_118_269} One end of a light inextensible string is attached to a particle $A$ of mass 2 kg . The other end of the string is attached to a second particle $B$ of mass 3 kg . Particle $A$ is in contact with a smooth plane inclined at $30 ^ { \circ }$ to the horizontal and particle $B$ is in contact with a rough horizontal plane. A second light inextensible string is attached to $B$. The other end of this second string is attached to a third particle $C$ of mass 4 kg . Particle $C$ is in contact with a smooth plane $\Pi$ inclined at an angle of $60 ^ { \circ }$ to the horizontal. Both strings are taut and pass over small smooth pulleys that are at the tops of the inclined planes. The parts of the strings from $A$ to the pulley, and from $C$ to the pulley, are parallel to lines of greatest slope of the corresponding planes (see diagram). The coefficient of friction between $B$ and the horizontal plane is $\mu$. The system is released from rest and in the subsequent motion $C$ moves down $\Pi$ with acceleration $a \mathrm {~ms} ^ { - 2 }$.

By considering an equation involving $\mu , a$ and $g$ show that $a < \frac { 1 } { 9 } g ( 2 \sqrt { 3 } - 1 )$.
Given that $a = \frac { 1 } { 9 } g$, determine the magnitude of the contact force between $B$ and the horizontal plane. Give your answer correct to $\mathbf { 3 }$ significant figures.
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SPS SPS SM Mechanics 2023 January Q1

10 seconds after passing a warning signal, a train is travelling at $18 m s ^ { - 1 }$ and has gone 215 m beyond the signal. Find the acceleration (assumed to be constant) of the train during the 10 seconds and its velocity as it passed the signal.

\section*{BLANK PAGE FOR WORKING}

SPS SPS SM Mechanics 2023 January Q2

2. A particle of mass $m$ is placed on a rough inclined plane.
The plane makes an angle $\theta$ with the horizontal.
The coefficient of friction between the particle and the plane is $\mu$ where $\mu < \tan \theta$. The particle is released from rest and accelerates down the plane.

Draw a fully labelled diagram to show the forces acting on the particle.
Find an expression in terms of $g , \theta$ and $\mu$ for the acceleration of the particle.
Explain what would happen to the particle if $\mu > \tan \theta$. \section*{BLANK PAGE FOR WORKING}

SPS SPS SM Mechanics 2023 January Q3

A parcel $P$ of weight 50 N is being held in equilibrium by two light, inextensible strings $A P$ and $B P$. The string $A P$ is attached to a wall at $A$, and string $B P$ passes over a smooth pulley which is at the same height as $A$, as shown in the diagram.

When the tension in $B P$ is 40 N , the strings are at right angles to each other.
\includegraphics[max width=\textwidth, alt={}, center]{4109fba0-077e-472b-b37f-7ac2e45aacc7-08_531_768_479_699}

Find the tension in string $A P$.
Explain why the parcel can never be in equilibrium, with both strings horizontal. \section*{BLANK PAGE FOR WORKING}

SPS SPS SM Mechanics 2023 January Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4109fba0-077e-472b-b37f-7ac2e45aacc7-10_680_1218_141_466} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A small ball, $P$, of mass 0.8 kg , is held at rest on a smooth horizontal table and is attached to one end of a thin rope. The rope passes over a pulley that is fixed at the edge of the table.
The other end of the rope is attached to another small ball, $Q$, of mass 0.6 kg , that hangs freely below the pulley. Ball $P$ is released from rest, with the rope taut, with $P$ at a distance of 1.5 m from the pulley and with $Q$ at a height of 0.4 m above the horizontal floor, as shown in Figure 1. Ball $Q$ descends, hits the floor and does not rebound.
The balls are modelled as particles, the rope as a light and inextensible string and the pulley as small and smooth. Using this model,

show that the acceleration of $Q$, as it falls, is $4.2 \mathrm {~ms} ^ { - 2 }$
find the time taken by $P$ to hit the pulley from the instant when $P$ is released.
State one limitation of the model that will affect the accuracy of your answer to part (a). \section*{BLANK PAGE FOR WORKING}

SPS SPS SM Mechanics 2023 January Q5

At time $t$ seconds, where $0 \leq t \leq T$, a particle, $P$, moves so that its velocity $v m s ^ { - 1 }$ is given by

$$v = 7.2 t - 0.45 t ^ { 2 }$$ When $t = 0$ the particle is sitting stationary at a displacement of $\mathrm { d } m$ from a point O .
The particle's acceleration is zero when $t = T$.

Find the value of $T$. For $t \geq T$, the particle moves with a velocity $v = 48 - 2.4 t m s ^ { - 1 }$.
Find the time when $P$ is at its maximum displacement from O . The particle passes through the point O when $t = 38$.
Find $d$. \section*{BLANK PAGE FOR WORKING}

Questions — SPS SPS SM Mechanics (39 questions)

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