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CAIE M1 2014 June Q4

Easy -3.0

4 A at $t$ vta a th
Th at $v$ avt
a $t$ av hta
That $v$ avt $t$ tht th $t$ $h$ at $t$ th tat $t$ th at $h$ $t$ th th tta
ta tav a
th tta t tak

CAIE M1 2014 June Q5

Moderate -0.5

5 \includegraphics[max width=\textwidth, alt={}, center]{9950eb77-25f9-4275-88ab-8dce9fb07cec-2_135_1351_1928_415} A a a k tat tat
th $a$ th hta th $t \quad$ taht $a$ hta aa $h$ $a$ ath tat th $a t \quad T h \quad$ th $a$ a ha tav a ta

th a $t$ t a tta $t$
$h$ that Th $t$ at a hht av thv
tha $v$ th $v$ th $t \quad t t a k t \quad t$ th va hh
th va hh
taht
Th $h t h$ $a$ $a$ $h h$

CAIE M1 2014 June Q6

Moderate -0.5

6 \includegraphics[max width=\textwidth, alt={}, center]{9950eb77-25f9-4275-88ab-8dce9fb07cec-3_492_556_260_792} Pat aka akaattahtt a ht xt thh $a v a x$ th Tht $h$ at $t$ th th $t$ tat a $t$ taht at vta Bt $h$ ataatahht $\quad$ avth $T h t a$ t $a$ at $h$ th at $a$ th $t$ ak Th at $t$ ah th th $t$ t

th aat
a th ta tav
th t ak \includegraphics[max width=\textwidth, alt={}, center]{9950eb77-25f9-4275-88ab-8dce9fb07cec-3_668_1086_1336_518} Th vtt ah $h$ th $t$ at
t t ht th Th
a th t tak $t$ ah $t$ att $a a h$

7		$a$	tav a a taht a Bth t a thh				\multirow[t]{4}{*}{$a$ av tat th}
	$T t$ ta at				tat ta at at av	$C t t h a h$
a a th th tat aat
	(i)	$h$ that th ta	$a$	at
	$C t$		$t$	at $t$	at av		h
	$a$
	(ii)		$a$ th $a x$		h ha ah
	$C t$ tav		$t \quad$ a ta at th h ha ah at $t$ th tat aat					Ct
	(iii)		th t tak th t t tav		$t$

CAIE M1 2014 June Q1

1 A a a tav a a tat $t$ tat a a $t$ a a T tat tat

$t$ va $v$ tat $t$ a aat at a tat $t a$ at at a at
tat ta tavta tat 2 a att a aat ta taa A at
tat t ta t tat aat At at v
t ta t tat a at at t tat tat tat t
v At t at t tat at $a$
$t v a$
$t$ at $t$

CAIE M1 2014 June Q3

Moderate -0.5

3 \includegraphics[max width=\textwidth, alt={}, center]{cb3c4728-fd7f-48c0-bbd0-7ea97c6d50f2-2_522_633_1135_740}

$a$		vta av	A at	$a$
atta $t$		a atta $t$		a txt
$t \quad t$	ata at a ta	$t a a$
ta at at		aa Bt $t$ a tat a $t$
at ta	atta tt tt

\multirow[t]{3}{*}{4}

A at

$v$ a tat tat tat a $t$

\multirow{2}{*}{}

$t$ Ttat

\multirow{2}{*}{

}
	$t v$	a tatva		t tat aat			-
	tat	at $t$		At a $t$	$t v t$		- ${ } ^ { - }$	\multirow{4}{*}{}	\multirow{4}{*}{}
\multirow{3}{*}{}	(i) taat	\multirow{2}{*}{}	\multirow{2}{*}{atattat atat}		\multirow{2}{*}{}	tat t aat
	$a$
	(ii) tav

\includegraphics[max width=\textwidth, alt={}]{cb3c4728-fd7f-48c0-bbd0-7ea97c6d50f2-3_334_689_258_726}

\multirow{2}{*}{At xt a a atta at $t t T a v a t a$ at a a}	$a$ atta at $a a$		\multirow[t]{2}{*}{$a$}
	$x$ att t a	at $t$ at $t$ tt $t a$
$a$	aa Tttt			a $t$
$a$	a tatt tat $v$ a $t$
a v a ta	a a

t tat tt t
$t$ (a) $t$ avtata tta $t t$ (b) $t$ aat $t$ ta
tat

CAIE M1 2014 June Q6

6 Aat a a tatat avtat a ta attaatt $T$ a taa $t$ tata a $a t$ at a ta $t$ a $t T \quad$ tt a tat ta tt at

tat tattatva Tt $t$ ta a $t$ xt $t$ a $t$ tat aat $T$ ta $t t$ a ta at av t tt t ta
tt tt
ta t ta a atta $t$ a at t tt t ta a t vta
t t at T at

CAIE M1 2014 June Q7

Easy -1.2

7 \includegraphics[max width=\textwidth, alt={}, center]{cb3c4728-fd7f-48c0-bbd0-7ea97c6d50f2-4_323_1008_274_566}

A $t$ xt $t \quad t \quad a$ at tv atta $t$ t At at tt Tat

\multirow[b]{4}{*}{$a$}

\multirow{6}{*}{$a$}

\multirow{4}{*}{at $t$ vta}

a a t t tat $t v t$ tat aat at

CAIE M1 2014 June Q1

5 marks Moderate -0.8

1 \includegraphics[max width=\textwidth, alt={}, center]{139371b7-e142-4ed6-bff3-3ca4c32b9c6b-2_426_424_258_863} A block $B$ of mass 7 kg is at rest on rough horizontal ground. A force of magnitude $X \mathrm {~N}$ acts on $B$ at an angle of $15 ^ { \circ }$ to the upward vertical (see diagram).

Given that $B$ is in equilibrium find, in terms of $X$, the normal component of the force exerted on $B$ by the ground.
The coefficient of friction between $B$ and the ground is 0.4 . Find the value of $X$ for which $B$ is in limiting equilibrium.

Show that the greatest height above the ground that the ball reaches after hitting the ground is 1.8 m .
Find the time taken for the ball's motion from its release until reaching this greatest height.

CAIE M1 2014 June Q5

8 marks Standard +0.3

5 A lorry of mass 16000 kg travels at constant speed from the bottom, $O$, to the top, $A$, of a straight hill. The distance $O A$ is 1200 m and $A$ is 18 m above the level of $O$. The driving force of the lorry is constant and equal to 4500 N .

Find the work done against the resistance to the motion of the lorry. On reaching $A$ the lorry continues along a straight horizontal road against a constant resistance of 2000 N . The driving force of the lorry is not now constant, and the speed of the lorry increases from $9 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $A$ to $21 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at the point $B$ on the road. The distance $A B$ is 2400 m .
Use an energy method to find $F$, where $F \mathrm {~N}$ is the average value of the driving force of the lorry while moving from $A$ to $B$.
Given that the driving force at $A$ is 1280 N greater than $F \mathrm {~N}$ and that the driving force at $B$ is 1280 N less than $F \mathrm {~N}$, show that the power developed by the lorry's engine is the same at $B$ as it is at $A$.

CAIE M1 2014 June Q6

10 marks Moderate -0.3

6 A particle starts from rest at a point $O$ and moves in a horizontal straight line. The velocity of the particle is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t \mathrm {~s}$ after leaving $O$. For $0 \leqslant t < 60$, the velocity is given by $$v = 0.05 t - 0.0005 t ^ { 2 }$$ The particle hits a wall at the instant when $t = 60$, and reverses the direction of its motion. The particle subsequently comes to rest at the point $A$ when $t = 100$, and for $60 < t \leqslant 100$ the velocity is given by $$v = 0.025 t - 2.5$$

Find the velocity of the particle immediately before it hits the wall, and its velocity immediately after its hits the wall.
Find the total distance travelled by the particle.
Find the maximum speed of the particle and sketch the particle's velocity-time graph for $0 \leqslant t \leqslant 100$, showing the value of $t$ for which the speed is greatest. \section*{[Question 7 is printed on the next page.]}

CAIE M1 2014 June Q7

9 marks Standard +0.3

7 \includegraphics[max width=\textwidth, alt={}, center]{139371b7-e142-4ed6-bff3-3ca4c32b9c6b-4_342_1257_255_445} A smooth inclined plane of length 160 cm is fixed with one end at a height of 40 cm above the other end, which is on horizontal ground. Particles $P$ and $Q$, of masses 0.76 kg and 0.49 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the top of the plane. Particle $P$ is held at rest on the same line of greatest slope as the pulley and $Q$ hangs vertically below the pulley at a height of 30 cm above the ground (see diagram). $P$ is released from rest. It starts to move up the plane and does not reach the pulley. Find

the acceleration of the particles and the tension in the string before $Q$ reaches the ground,
the speed with which $Q$ reaches the ground,
the total distance travelled by $P$ before it comes to instantaneous rest.

CAIE M1 2015 June Q1

4 marks Moderate -0.8

1 A block $B$ of mass 2.7 kg is pulled at constant speed along a straight line on a rough horizontal floor. The pulling force has magnitude 25 N and acts at an angle of $\theta$ above the horizontal. The normal component of the contact force acting on $B$ has magnitude 20 N .

Show that $\sin \theta = 0.28$.
Find the work done by the pulling force in moving the block a distance of 5 m .

CAIE M1 2015 June Q2

5 marks Moderate -0.3

2 \includegraphics[max width=\textwidth, alt={}, center]{f4f2996b-5382-4b0d-9804-b5f5945946b3-2_636_519_664_813} Three horizontal forces of magnitudes $F \mathrm {~N} , 63 \mathrm {~N}$ and 25 N act at $O$, the origin of the $x$-axis and $y$-axis. The forces are in equilibrium. The force of magnitude $F \mathrm {~N}$ makes an angle $\theta$ anticlockwise with the positive $x$-axis. The force of magnitude 63 N acts along the negative $y$-axis. The force of magnitude 25 N acts at $\tan ^ { - 1 } 0.75$ clockwise from the negative $x$-axis (see diagram). Find the value of $F$ and the value of $\tan \theta$.

CAIE M1 2015 June Q3

5 marks Standard +0.3

3 A block of weight 6.1 N slides down a slope inclined at $\tan ^ { - 1 } \left( \frac { 11 } { 60 } \right)$ to the horizontal. The coefficient of friction between the block and the slope is $\frac { 1 } { 4 }$. The block passes through a point $A$ with speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find how far the block moves from $A$ before it comes to rest.

CAIE M1 2015 June Q4

6 marks Moderate -0.3

4 A lorry of mass 14000 kg moves along a road starting from rest at a point $O$. It reaches a point $A$, and then continues to a point $B$ which it reaches with a speed of $24 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The part $O A$ of the road is straight and horizontal and has length 400 m . The part $A B$ of the road is straight and is inclined downwards at an angle of $\theta ^ { \circ }$ to the horizontal and has length 300 m .

For the motion from $O$ to $B$, find the gain in kinetic energy of the lorry and express its loss in potential energy in terms of $\theta$. The resistance to the motion of the lorry is 4800 N and the work done by the driving force of the lorry from $O$ to $B$ is 5000 kJ .
Find the value of $\theta$.

CAIE M1 2015 June Q5

8 marks Standard +0.8

5 A cyclist and her bicycle have a total mass of 84 kg . She works at a constant rate of $P \mathrm {~W}$ while moving on a straight road which is inclined to the horizontal at an angle $\theta$, where $\sin \theta = 0.1$. When moving uphill, the cyclist's acceleration is $1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ at an instant when her speed is $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When moving downhill, the cyclist's acceleration is $1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ at an instant when her speed is $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The resistance to the cyclist's motion, whether the cyclist is moving uphill or downhill, is $R \mathrm {~N}$. Find the values of $P$ and $R$.

CAIE M1 2015 June Q6

10 marks Standard +0.8

6 Two particles $A$ and $B$ start to move at the same instant from a point $O$. The particles move in the same direction along the same straight line. The acceleration of $A$ at time $t \mathrm {~s}$ after starting to move is $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$, where $a = 0.05 - 0.0002 t$.

Find A's velocity when $t = 200$ and when $t = 500$. $B$ moves with constant acceleration for the first 200 s and has the same velocity as $A$ when $t = 200 . B$ moves with constant retardation from $t = 200$ to $t = 500$ and has the same velocity as $A$ when $t = 500$.
Find the distance between $A$ and $B$ when $t = 500$.

CAIE M1 2015 June Q7

12 marks Standard +0.8

7 \includegraphics[max width=\textwidth, alt={}, center]{f4f2996b-5382-4b0d-9804-b5f5945946b3-3_376_1052_1171_548} Particles $A$ and $B$, of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string. Particle $A$ is held at rest on a rough horizontal table with the string passing over a smooth pulley fixed at the edge of the table. The coefficient of friction between $A$ and the table is 0.2 . Particle $B$ hangs vertically below the pulley at a height of 0.5 m above the floor (see diagram). The system is released from rest and 0.25 s later the string breaks. A does not reach the pulley in the subsequent motion. Find

the speed of $B$ immediately before it hits the floor,
the total distance travelled by $A$.

CAIE M1 2015 June Q1

4 marks Moderate -0.3

1 One end of a light inextensible string is attached to a block. The string makes an angle of $60 ^ { \circ }$ above the horizontal and is used to pull the block in a straight line on a horizontal floor with acceleration $0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The tension in the string is 8 N . The block starts to move with speed $0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. For the first 5 s of the block's motion, find

the distance travelled,
the work done by the tension in the string.

CAIE M1 2015 June Q2

5 marks Moderate -0.3

2 The total mass of a cyclist and his cycle is 80 kg . The resistance to motion is zero.

The cyclist moves along a horizontal straight road working at a constant rate of $P \mathrm {~W}$. Find the value of $P$ given that the cyclist's speed is $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when his acceleration is $1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
The cyclist moves up a straight hill inclined at an angle $\alpha$, where $\sin \alpha = 0.035$. Find the acceleration of the cyclist at an instant when he is working at a rate of 450 W and has speed $3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

CAIE M1 2015 June Q3

5 marks Moderate -0.3

3 A plane is inclined at an angle of $\sin ^ { - 1 } \left( \frac { 1 } { 8 } \right)$ to the horizontal. $A$ and $B$ are two points on the same line of greatest slope with $A$ higher than $B$. The distance $A B$ is 12 m . A small object $P$ of mass 8 kg is released from rest at $A$ and slides down the plane, passing through $B$ with speed $4.5 \mathrm {~ms} ^ { - 1 }$. For the motion of $P$ from $A$ to $B$, find

the increase in kinetic energy of $P$ and the decrease in potential energy of $P$,
the magnitude of the constant resisting force that opposes the motion of $P$.

CAIE M1 2015 June Q4

7 marks Standard +0.3

4 A particle $P$ moves in a straight line. At time $t$ seconds after starting from rest at the point $O$ on the line, the acceleration of $P$ is $a \mathrm {~m} \mathrm {~s} ^ { - 2 }$, where $a = 0.075 t ^ { 2 } - 1.5 t + 5$.

Find an expression for the displacement of $P$ from $O$ in terms of $t$.
Hence find the time taken for $P$ to return to the point $O$.

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7		\(a\)	tav a a taht a Bth t a thh				\multirow[t]{4}{*}{\(a\) av tat th}
	\(T t\) ta at				tat ta at at av	\(C t t h a h\)
a a th th tat aat
	(i)	\(h\) that th ta	\(a\)	at
	\(C t\)		\(t\)	at \(t\)	at av		h
	\(a\)
	(ii)		\(a\) th \(a x\)		h ha ah
	\(C t\) tav		\(t \quad\) a ta at th h ha ah at \(t\) th tat aat					Ct
	(iii)		th t tak th t t tav		\(t\)

tat	t ta	t tat aat		At at	v
t ta	t tat		a at	at t tat tat	tat t
v At t	at t tat at	\(a\)

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CAIE M1 2014 June Q4

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