Questions - CAIE Further Paper 3

Derive the equation of the trajectory of $P$ in the form $$\mathrm { y } = \mathrm { x } \tan \alpha - \frac { \mathrm { gx } ^ { 2 } } { 2 \mathrm { u } ^ { 2 } } \sec ^ { 2 } \alpha$$ During its flight, $P$ must clear an obstacle of height $h \mathrm {~m}$ that is at a horizontal distance of 32 m from the point of projection. When $u = 40 \sqrt { 2 } \mathrm {~ms} ^ { - 1 } , P$ just clears the obstacle. When $u = 40 \mathrm {~ms} ^ { - 1 } , P$ only achieves $80 \%$ of the height required to clear the obstacle.
Find the two possible values of $h$.

CAIE Further Paper 3 2023 November Q7

7
\includegraphics[max width=\textwidth, alt={}, center]{0270d51a-e252-46d3-8c97-7f71ba91fa65-12_618_835_255_616} A particle $P$ of mass $m$ is attached to one end of a light rod of length $3 a$. The other end of the rod is able to pivot smoothly about the fixed point $A$. The particle is also attached to one end of a light spring of natural length $a$ and modulus of elasticity $k m g$. The other end of the spring is attached to a fixed point $B$. The points $A$ and $B$ are in a horizontal line, a distance $5 a$ apart, and these two points and the rod are in a vertical plane. Initially, $P$ is held in equilibrium by a vertical force $F$ with the stretched length of the spring equal to $4 a$ (see diagram). The particle is released from rest in this position and has a speed of $\frac { 6 } { 5 } \sqrt { 2 a g }$ when the rod becomes horizontal.

Find the value of $k$.
Find $F$ in terms of $m$ and $g$.
Find, in terms of $m$ and $g$, the tension in the rod immediately before it is released.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

CAIE Further Paper 3 2023 November Q2

2 A ball of mass 2 kg is projected vertically downwards with speed $5 \mathrm {~ms} ^ { - 1 }$ through a liquid. At time $t \mathrm {~s}$ after projection, the velocity of the ball is $v \mathrm {~ms} ^ { - 1 }$ and its displacement from its starting point is $x \mathrm {~m}$. The forces acting on the ball are its weight and a resistive force of magnitude $0.2 v ^ { 2 } \mathrm {~N}$.

Find an expression for $v$ in terms of $t$.
Deduce what happens to $v$ for large values of $t$.
\includegraphics[max width=\textwidth, alt={}, center]{e7091f6c-af72-49f3-b825-cdce9fb2c06f-06_803_652_251_703} A uniform square lamina of side $2 a$ and weight $W$ is suspended from a light inextensible string attached to the midpoint $E$ of the side $A B$. The other end of the string is attached to a fixed point $P$ on a rough vertical wall. The vertex $B$ of the lamina is in contact with the wall. The string $E P$ is perpendicular to the side $A B$ and makes an angle $\theta$ with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is $\frac { 1 } { 2 }$. Given that the vertex $B$ is about to slip up the wall, find the value of $\tan \theta$.
\includegraphics[max width=\textwidth, alt={}, center]{e7091f6c-af72-49f3-b825-cdce9fb2c06f-08_581_576_269_731} A light elastic string has natural length $8 a$ and modulus of elasticity $5 m g$. A particle $P$ of mass $m$ is attached to the midpoint of the string. The ends of the string are attached to points $A$ and $B$ which are a distance $12 a$ apart on a smooth horizontal table. The particle $P$ is held on the table so that $A P = B P = L$ (see diagram). The particle $P$ is released from rest. When $P$ is at the midpoint of $A B$ it has speed $\sqrt { 80 a g }$.
Find $L$ in terms of $a$.
Find the initial acceleration of $P$ in terms of $g$.

CAIE Further Paper 3 2024 November Q1

1 A particle $P$ is projected with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\tan ^ { - 1 } 2$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. When $P$ has travelled a distance 56 m horizontally from $O$, it is at a vertical height $H \mathrm {~m}$ above the plane. When $P$ has travelled a distance 84 m horizontally from $O$, it is at a vertical height $\frac { 1 } { 2 } H \mathrm {~m}$ above the plane. Find, in either order, the value of $u$ and the value of $H$.
\includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-02_2718_38_106_2009}

CAIE Further Paper 3 2024 November Q2

2 A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle $P$ is held at the point $A$ with the string taut. It is given that $O A$ makes an angle $\theta$ with the downward vertical through $O$, where $\tan \theta = \frac { 3 } { 4 }$. The particle $P$ is projected perpendicular to $O A$ in an upwards direction with speed $\sqrt { 5 a g }$, and it starts to move along a circular path in a vertical plane. When $P$ is at the point $B$, where angle $A O B$ is a right angle, the tension in the string is $T$. Find $T$ in terms of $m$ and $g$.

CAIE Further Paper 3 2024 November Q3

3 A particle $P$ of mass $m \mathrm {~kg}$ is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity $2 m g \mathrm {~N}$. The other end of the string is attached to a fixed point $O$. The particle $P$ hangs in equilibrium vertically below $O$. The particle $P$ is pulled down vertically a distance $d$ m below its equilibrium position and released from rest.

Given that the particle just reaches $O$ in the subsequent motion, find the value of $d$.
\includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-04_2713_31_111_2017}
\includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-05_2725_35_99_20}
Hence find the speed of $P$ when it is 2 m below $O$.
\includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-06_785_729_255_708} An object is formed by removing a cylinder of radius $\frac { 2 } { 3 } a$ and height $k h ( k < 1 )$ from a uniform solid cylinder of radius $a$ and height $h$. The vertical axes of symmetry of the two cylinders coincide. The upper faces of the two cylinders are in the same plane as each other. The points $A$ and $B$ are the opposite ends of a diameter of the upper face of the object (see diagram).
Find, in terms of $h$ and $k$, the distance of the centre of mass of the object from $A B$.

When the object is suspended from $A$, the angle between $A B$ and the vertical is $\theta$, where $\tan \theta = \frac { 3 } { 2 }$.
Given that $h = \frac { 8 } { 3 } a$, find the possible values of $k$.

CAIE Further Paper 3 2024 November Q5

5 A particle $P$ of mass 2 kg moving on a horizontal straight line has displacement $x \mathrm {~m}$ from a fixed point $O$ on the line and velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t$ s. The only horizontal force acting on $P$ is a variable force $F \mathrm {~N}$ which can be expressed as a function of $t$. It is given that $$\frac { v } { x } = \frac { 3 - t } { 1 + t }$$ and when $t = 0 , x = 5$.

Find an expression for $x$ in terms of $t$.
\includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-08_2718_38_106_2009}
\includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-09_2723_33_99_21}
Find the magnitude of $F$ when $t = 3$.
\includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-10_559_1257_255_445} A particle $P$ of mass 0.05 kg is attached to one end of a light inextensible string of length 1 m . The other end of the string is attached to a fixed point $O$. A particle $Q$ of mass 0.04 kg is attached to one end of a second light inextensible string. The other end of this string is attached to $P$. The particle $P$ moves in a horizontal circle of radius 0.8 m with angular speed $\omega \operatorname { rad~s } ^ { - 1 }$. The particle $Q$ moves in a horizontal circle of radius 1.4 m also with angular speed $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$. The centres of the circles are vertically below $O$, and $O , P$ and $Q$ are always in the same vertical plane. The strings $O P$ and $P Q$ remain at constant angles $\alpha$ and $\beta$ respectively to the vertical (see diagram).
Find the tension in the string $O P$.
\includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-10_2716_38_109_2012}
\includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-11_2725_35_99_20}
Find the value of $\omega$.
Find the value of $\beta$.

CAIE Further Paper 3 2024 November Q7

7 A particle $P$ is projected with speed $u$ at an angle $\tan ^ { - 1 } \left( \frac { 4 } { 3 } \right)$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. When $P$ is moving horizontally, it strikes a smooth inclined plane at the point $A$. This plane is inclined to the horizontal at an angle $\alpha$, and the line of greatest slope through $A$ lies in the vertical plane through $O$ and $A$. As a result of the impact, $P$ moves vertically upwards. The coefficient of restitution between $P$ and the inclined plane is $e$.

Show that $e \tan ^ { 2 } \alpha = 1$.
\includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-12_2716_36_106_2014} In its subsequent motion, the greatest height reached by $P$ above $A$ is $\frac { 3 } { 16 }$ of the vertical height of $A$ above the horizontal plane.
Find the value of $e$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.
\includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-14_2715_33_109_2012}

CAIE Further Paper 3 2024 November Q1

1 A particle of mass 2 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 100 N . The other end of the string is attached to a fixed point $O$ on a smooth horizontal surface. The particle is moving in a horizontal circle about $O$ with the string taut and with constant angular speed 5 radians per second. Find the extension of the string.
\includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-02_2717_33_109_2014}

CAIE Further Paper 3 2024 November Q2

2 A particle $P$ of mass $m$ is attached to one end of a light elastic spring of natural length $a$ and modulus of elasticity 5 mg . The other end of the spring is attached to a fixed point $O$. The spring hangs vertically with $P$ below $O$. The particle $P$ is pulled down vertically and released from rest when the length of the spring is $\frac { 3 } { 2 } a$. Find the distance of $P$ below $O$ when $P$ first comes to instantaneous rest.

CAIE Further Paper 3 2024 November Q3

3
\includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-04_348_828_251_621}
\includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-04_2717_35_110_2012} The diagram shows two identical smooth uniform spheres $A$ and $B$ of equal radii and each of mass $m$. The two spheres are moving on a smooth horizontal surface when they collide with speeds $2 u$ and $3 u$ respectively. Immediately before the collision, $A$ 's direction of motion makes an angle $\theta$ with the line of centres and $B$ 's direction of motion is perpendicular to that of $A$. After the collision, $B$ moves perpendicular to the line of centres. The coefficient of restitution between the spheres is $\frac { 1 } { 3 }$.

Find the value of $\tan \theta$.
\includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-05_2723_33_99_21}
Find the total loss of kinetic energy as a result of the collision.
Find, in degrees, the angle through which the direction of motion of $A$ is deflected as a result of the collision.
\includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-06_776_785_255_680} The end $A$ of a uniform rod $A B$ of length $6 a$ and weight $W$ is in contact with a rough vertical wall. One end of a light inextensible string of length $3 a$ is attached to the midpoint $C$ of the rod. The other end of the string is attached to a point $D$ on the wall, vertically above $A$. The rod is in equilibrium when the angle between the rod and the wall is $\theta$, where $\tan \theta = \frac { 3 } { 2 }$. A particle of weight $W$ is attached to the point $E$ on the rod, where the distance $A E$ is equal to $k a ( 3 < k < 6 )$ (see diagram). The rod and the string are in a vertical plane perpendicular to the wall. The coefficient of friction between the rod and the wall is $\frac { 1 } { 3 }$. The rod is about to slip down the wall.
Find the value of $k$.
Find, in terms of $W$, the magnitude of the frictional force between the rod and the wall.

CAIE Further Paper 3 2024 November Q5

5 A particle $P$ is projected from a point $O$ on horizontal ground with speed $u$ at an angle $\theta$ above the horizontal, where $\tan \theta = \frac { 1 } { 3 }$. The particle $P$ moves freely under gravity and passes through the point with coordinates $\left( 3 a , \frac { 4 } { 5 } a \right)$ relative to horizontal and vertical axes through $O$ in the plane of the motion.

Use the equation of the trajectory to show that $u ^ { 2 } = 25 a g$.
\includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-09_2725_35_99_20} At the instant when $P$ is moving horizontally, a particle $Q$ is projected from $O$ with speed $V$ at an angle $\alpha$ above the horizontal. The particles $P$ and $Q$ reach the ground at the same point and at the same time.
Express $V ^ { 2 }$ in the form $k a g$, where $k$ is a rational number.
\includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-10_506_803_255_630} A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle $P$ is held with the string taut and the string makes an angle $\theta$ with the downward vertical through $O$. The particle $P$ is projected at right angles to the string with speed $\frac { 1 } { 3 } \sqrt { 10 a g }$ and begins to move downwards along a circular path. When the string is vertical, it strikes a small smooth peg at the point $A$ which is vertically below $O$. The circular path and the point $A$ are in the same vertical plane. After the string strikes the peg, the particle $P$ begins to move in a vertical circle with centre $A$. When the string makes an angle $\theta$ with the upward vertical through $A$ the string becomes slack (see diagram). The distance of $A$ below $O$ is $\frac { 5 } { 9 } a$.
Find the value of $\cos \theta$.
\includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-11_2725_35_99_20}
Find the ratio of the tensions in the string immediately before and immediately after it strikes the peg.
$7 \quad$ A particle $P$ of mass $m \mathrm {~kg}$ is held at rest at a point $O$ and released so that it moves vertically under gravity against a resistive force of magnitude $0.1 m v ^ { 2 } \mathrm {~N}$, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the velocity of $P$ at time $t \mathrm {~s}$.
Find an expression for $v$ in terms of $t$.
The displacement of $P$ from $O$ at time $t \mathrm {~s}$ is $x \mathrm {~m}$.
Find an expression for $v ^ { 2 }$ in terms of $x$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.
\includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-14_2716_31_106_2016}

CAIE Further Paper 3 2024 November Q1

1 A particle $P$ is projected with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle $\tan ^ { - 1 } 2$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. When $P$ has travelled a distance 56 m horizontally from $O$, it is at a vertical height $H \mathrm {~m}$ above the plane. When $P$ has travelled a distance 84 m horizontally from $O$, it is at a vertical height $\frac { 1 } { 2 } H \mathrm {~m}$ above the plane. Find, in either order, the value of $u$ and the value of $H$.
\includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-02_2718_38_106_2009}

CAIE Further Paper 3 2024 November Q3

Given that the particle just reaches $O$ in the subsequent motion, find the value of $d$.
\includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-04_2717_33_109_2014}
\includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-05_2725_35_99_20}
Hence find the speed of $P$ when it is 2 m below $O$.
\includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-06_785_729_255_708} An object is formed by removing a cylinder of radius $\frac { 2 } { 3 } a$ and height $k h ( k < 1 )$ from a uniform solid cylinder of radius $a$ and height $h$. The vertical axes of symmetry of the two cylinders coincide. The upper faces of the two cylinders are in the same plane as each other. The points $A$ and $B$ are the opposite ends of a diameter of the upper face of the object (see diagram).
Find, in terms of $h$ and $k$, the distance of the centre of mass of the object from $A B$.

When the object is suspended from $A$, the angle between $A B$ and the vertical is $\theta$, where $\tan \theta = \frac { 3 } { 2 }$.
Given that $h = \frac { 8 } { 3 } a$, find the possible values of $k$.

CAIE Further Paper 3 2024 November Q5

Find an expression for $x$ in terms of $t$.
\includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-09_2725_35_99_20}
Find the magnitude of $F$ when $t = 3$.
\includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-10_559_1257_255_445} A particle $P$ of mass 0.05 kg is attached to one end of a light inextensible string of length 1 m . The other end of the string is attached to a fixed point $O$. A particle $Q$ of mass 0.04 kg is attached to one end of a second light inextensible string. The other end of this string is attached to $P$. The particle $P$ moves in a horizontal circle of radius 0.8 m with angular speed $\omega \operatorname { rad~s } ^ { - 1 }$. The particle $Q$ moves in a horizontal circle of radius 1.4 m also with angular speed $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$. The centres of the circles are vertically below $O$, and $O , P$ and $Q$ are always in the same vertical plane. The strings $O P$ and $P Q$ remain at constant angles $\alpha$ and $\beta$ respectively to the vertical (see diagram).
Find the tension in the string $O P$.
\includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-10_2718_42_107_2007}
\includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-11_2725_35_99_20}
Find the value of $\omega$.
Find the value of $\beta$.

CAIE Further Paper 3 2024 November Q7

Show that $e \tan ^ { 2 } \alpha = 1$.
\includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-12_2713_33_111_2017} In its subsequent motion, the greatest height reached by $P$ above $A$ is $\frac { 3 } { 16 }$ of the vertical height of $A$ above the horizontal plane.
Find the value of $e$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.
\includegraphics[max width=\textwidth, alt={}, center]{3cec9ccf-fb6a-4df3-8dfe-be8b092d3dd2-14_2715_33_109_2012}

CAIE Further Paper 3 2020 Specimen Q1

4 marks

1 A ch ld's ty co ists

CAIE Further Paper 3 2020 Specimen Q2

2 A lig elastic strig hsa tn al leg h $a$ ad md B 6 elasticity 2 mg . Or ed 6 th strig is attach d œ fiæ $\Phi \quad n A$. Tb ob rend te strig s attach d œ $\mathbf { P }$ rticle 6 mass $2 m$.

Fid in terms $6 a$, th ex en in 6 th strig wh $n$th $\boldsymbol { \rho }$ rticle $\mathbf { h }$ g freely in eq lib im b low $A$.
Th $\mathbf { p }$ rticle is released rm rest at $A$. Fid in terms $\mathbf { 6 } \quad a$, th $\dot { \mathbf { d } }$ stance

CAIE Further Paper 3 2020 Specimen Q3

3 A $\boldsymbol { \rho }$ rticle $P 6$ mass $m$ g falls frm rest $\mathbf { d } \quad \mathrm { rg }$ avty. The re is a resistiv fo ce 6 mag td $m k v ^ { 2 } \mathrm {~N}$, wh re $v \mathrm {~ms} ^ { - 1 }$ is th se e $\boldsymbol { C }$ after it $\mathbf { h }$ s fallera id stan e $x \mathrm {~m}$ ad $k$ is a $\mathbf { p }$ itie co tan.

Bys b in ra ra p p iate d fferen ial eq tin ,s how th t $$v ^ { 2 } = \frac { g } { k } \left( 1 - \mathrm { e } ^ { - 2 k x } \right)$$ [] It is $\mathbf { N }$ it $\mathbf { h } \mathrm { t } k = 0.01$. The speed of $P \mathrm { w } \mathbf { h } \mathrm { n } x \mathbf { b } \mathrm {~cm}$ es larg ap $\mathbf { o } \mathrm { ch } \mathrm { s } V \mathrm {~ms} ^ { - 1 }$.
1. Fid $V$ correct to 2 decimal places.
2. Hen e fidw far $P \mathbf { b }$ s fallenw $\mathbf { b }$ n ts sp ed $\mathrm { s } \frac { 1 } { 2 } V \mathrm {~ms} ^ { - 1 }$.
  \includegraphics[max width=\textwidth, alt={}]{1e5941a9-14eb-441c-8d39-fd62695446ac-08_319_908_255_580}
  Two in fo m smo h sp res $A$ ad $B \mathbf { 6 }$ eq $l$ radili masses $m$ ad $2 m$ resp ctie ly. Sp re $B$ is at rest $\mathbf { n }$ a smo hb izn al sn face. Sp ere $A$ is mi $\mathbf { g } \mathbf { n }$ th sn face with sp ed $u$ at an ag e $\mathbf { b } \boldsymbol { B }$ to th lin 6 cen res $6 A$ and $B$ wh n it cb lid s with $B$ (see $\dot { \mathbf { d } }$ ag am). Th ce fficien

CAIE Further Paper 3 2020 Specimen Q5

5 A $\mathbf { p }$ rticle $P \mathbf { 6 }$ mass $m$ is attach d to e. a lig in k en ib e strig leg h $a$. Tb o b re. th strig s attach d or fiæ $\Phi \quad n O$.

\includegraphics[max width=\textwidth, alt={}, center]{1e5941a9-14eb-441c-8d39-fd62695446ac-10_314_768_397_651} Th $\boldsymbol { p }$ rticle $P$ mo sinab izo al circle with a co tan ag arsp ed $\omega$ with th strig in lie d at $\theta$ to $\mathrm { b } \quad \mathrm { m }$ ard rtical th $\mathrm { g } \quad O$ (see $\dot { \mathrm { d } } \mathrm { ag } \mathrm { am }$ ). Sw that $\omega ^ { 2 } = \frac { 2 g } { a }$.
Th $\mathbf { p }$ rticle w has at restad stan e $a$ rtically b low $O$. It isth $\mathrm { np } \dot { \mathrm { p } }$ ected b izb ally so th t it $\mathbf { b } \mathbf { g }$ s to m in a rtical circle with cen re $O$. Wh n th strig maks an ag e $\boldsymbol { \theta }$, with th d $\mathbb { w }$ ard rtical th $\mathbf { g } O$, th ang arse $\operatorname { d } P$ is $\sqrt { \frac { 2 g } { a } }$. Tb strig first $\mathbf { g }$ s slack when $O P$ mak s ara g e $\theta$ with b ard rtical th ob $O$. Fid b le 6 co $\theta$.

CAIE Further Paper 3 2020 Specimen Q6

6 A $\mathbf { p }$ rticle $P$ is $\mathrm { p } \dot { \mathbf { j } }$ ected with sp ed $u$ at an ag $\mathrm { e } \alpha$ ab tb $\mathbf { b }$ izn al frm $\mathrm { a } \dot { \mathbf { p } }$ n $O \mathbf { n }$ ab izb al p ae ad mo s freely d rg av ty. Th $\mathbf { b }$ izo al ad rtical d sp acemen s $\mathbf { 6 } P$ frm $O$ at a sb eq $n$ time $t$ are $\mathbf { d } \mathbf { n }$ edy $\quad x$ ad $y$ resp ctie ly.

Derie the eq tirib the trajecto y $P$ irt b fo m $$y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \sec ^ { 2 } \alpha$$
Tb g eatestb in $6 P$ ab th p au is o edy $H$. Wh $\mathrm { n } P$ is ata $\mathbf { b }$ ig $6 \frac { 3 } { 4 } H$, it $\mathbf { a }$ strac lled ab izn ald stan e $d$. Gie it $\mathbf { h } \tan \alpha = \Im$ id it erms $6 H$,tb twœ sibed le $\mathrm { s } \varnothing d$. If B e th follw ig lin dpg to cm p ete th an wer(s) to ay q stin (s), th q stin $\mathrm { m } \quad \mathbf { b } \quad \mathrm { r } ( \mathrm { s } )$ ms tb clearlys n n

CAIE Further Paper 3 2020 June Q4

Show that $\bar { x } = \frac { 400 - x ^ { 2 } } { 80 - 3 x }$ and find a corresponding expression for $\bar { y }$.
The shape $A B E F D$ is in equilibrium in a vertical plane with the edge $D F$ resting on a smooth horizontal surface.
Find the greatest possible value of $x$, giving your answer in the form $\mathrm { a } + \mathrm { b } \sqrt { 2 }$, where $a$ and $b$ are constants to be determined.

CAIE Further Paper 3 2023 June Q4

Show that $\mathrm { x } = \frac { 32 \mathrm { a } ^ { 2 } + 3 \mathrm { ad } } { 16 \mathrm { a } + 3 \mathrm {~d} }$ and find an expression, in terms of $a$ and $d$, for $\bar { y }$.
The object is placed on a rough plane which is inclined to the horizontal at an angle $\theta$ where $\sin \theta = \frac { 1 } { 6 }$. The object is in equilibrium with $C O$ horizontal, where $C O$ lies in a vertical plane through a line of greatest slope.
Find $d$ in terms of $a$.

CAIE Further Paper 3 2023 June Q3

Find the value of $e$.
Find the loss in the total kinetic energy of the spheres as a result of the collision.

CAIE Further Paper 3 2021 November Q3

Show that the distance of the centre of mass of the object from $A B$ is $\frac { 3 \mathrm { a } \left( 2 - \mathrm { k } ^ { 2 } \right) } { 2 ( 8 - 3 \mathrm { k } ) }$.
When the object is freely suspended from the point $A$, the line $A B$ makes an angle $\theta$ with the downward vertical, where $\tan \theta = \frac { 7 } { 18 }$.
Find the possible values of $k$.
\includegraphics[max width=\textwidth, alt={}, center]{e34abb4b-1c6c-4f39-836d-467ed18345eb-08_494_903_267_525} Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $\frac { 3 } { 2 } m$ respectively. The two spheres are each moving with speed $u$ on a horizontal surface when they collide. Immediately before the collision $A$ 's direction of motion is along the line of centres, and $B$ 's direction of motion makes an angle of $60 ^ { \circ }$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac { 2 } { 3 }$.
Find the angle through which the direction of motion of $B$ is deflected by the collision.
Find the loss in the total kinetic energy of the system as a result of the collision.

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