Questions - CAIE Further Paper 3

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CAIE Further Paper 3 2021 November Q7

8 marks Challenging +1.8

One end of a light inextensible string of length $a$ is attached to a fixed point $O$. The other end of the string is attached to a particle $P$ of mass $m$. The particle $P$ is held vertically below $O$ with the string taut and then projected horizontally. When the string makes an angle of $60°$ with the upward vertical, $P$ becomes detached from the string. In its subsequent motion, $P$ passes through the point $A$ which is a distance $a$ vertically above $O$.

The speed of $P$ when it becomes detached from the string is $V$. Use the equation of the trajectory of a projectile to find $V$ in terms of $a$ and $g$. [4]
Find, in terms of $m$ and $g$, the tension in the string immediately after $P$ is initially projected horizontally. [4]

Find an expression for $v$ in terms of $x$. [6]
State the value that the speed approaches for large values of $x$. [1]

CAIE Further Paper 3 2022 November Q5

9 marks Standard +0.3

A particle $P$ is projected with speed $u$ m s$^{-1}$ at an angle of $\theta$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of $P$ from $O$ at a subsequent time $t$ s are denoted by $x$ m and $y$ m respectively.

Show that the equation of the trajectory is given by $$y = x \tan \theta - \frac{gx^2}{2u^2}(1 + \tan^2 \theta).$$ [4]

In the subsequent motion $P$ passes through the point with coordinates $(30, 20)$.

Given that one possible value of $\tan \theta$ is $\frac{4}{3}$, find the other possible value of $\tan \theta$. [5]

CAIE Further Paper 3 2022 November Q6

9 marks Challenging +1.8

\includegraphics{figure_6} A light inextensible string is threaded through a fixed smooth ring $R$ which is at a height $h$ above a smooth horizontal surface. One end of the string is attached to a particle $A$ of mass $m$. The other end of the string is attached to a particle $B$ of mass $\frac{1}{2}m$. The particle $A$ moves in a horizontal circle on the surface. The particle $B$ hangs in equilibrium below the ring and above the surface (see diagram). When $A$ has constant angular speed $\omega$, the angle between $AR$ and $BR$ is $\theta$ and the normal reaction between $A$ and the surface is $N$. When $A$ has constant angular speed $\frac{3}{2}\omega$, the angle between $AR$ and $BR$ is $\alpha$ and the normal reaction between $A$ and the surface is $\frac{1}{2}N$.

Show that $\cos \theta = \frac{4}{9}\cos \alpha$. [5]
Find $N$ in terms of $m$ and $g$ and find the value of $\cos \alpha$. [4]

CAIE Further Paper 3 2022 November Q7

9 marks Challenging +1.8

\includegraphics{figure_7} Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $\frac{1}{2}m$ respectively. The two spheres are moving on a horizontal surface when they collide. Immediately before the collision, sphere $A$ is travelling with speed $u$ and its direction of motion makes an angle $\alpha$ with the line of centres. Sphere $B$ is travelling with speed $2u$ and its direction of motion makes an angle $\beta$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac{5}{8}$ and $\alpha + \beta = 90°$.

Find the component of the velocity of $B$ parallel to the line of centres after the collision, giving your answer in terms of $u$ and $\alpha$. [4]

The direction of motion of $B$ after the collision is parallel to the direction of motion of $A$ before the collision.

Find the value of $\tan \alpha$. [5]

CAIE Further Paper 3 2023 November Q1

7 marks Challenging +1.8

\includegraphics{figure_1} Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $2m$ respectively. The two spheres are moving with equal speeds $u$ on a smooth horizontal surface when they collide. Immediately before the collision, $A$'s direction of motion makes an angle of $60°$ with the line of centres, and $B$'s direction of motion makes an angle $\theta$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $e$. After the collision, the component of the velocity of $A$ along the line of centres is $v$ and $B$ moves perpendicular to the line of centres. Sphere $A$ now has twice as much kinetic energy as sphere $B$.

Show that $v = \frac{1}{2}u(4\cos\theta - 1)$. [1]
Find the value of $\cos\theta$. [4]
Find the value of $e$. [2]

CAIE Further Paper 3 2023 November Q2

7 marks Challenging +1.2

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

Find an expression for $v$ in terms of $t$. [6]
Deduce what happens to $v$ for large values of $t$. [1]

CAIE Further Paper 3 2023 November Q3

8 marks Challenging +1.8

\includegraphics{figure_3} A uniform square lamina of side $2a$ and weight $W$ is suspended from a light inextensible string attached to the midpoint $E$ of the side $AB$. 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 $EP$ is perpendicular to the side $AB$ 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 $\tan\theta$. [8]

CAIE Further Paper 3 2023 November Q4

8 marks Challenging +1.2

\includegraphics{figure_4} A light elastic string has natural length $8a$ and modulus of elasticity $5mg$. 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 $12a$ apart on a smooth horizontal table. The particle $P$ is held on the table so that $AP = BP = L$ (see diagram). The particle $P$ is released from rest. When $P$ is at the midpoint of $AB$ it has speed $\sqrt{80ag}$.

Find $L$ in terms of $a$. [5]
Find the initial acceleration of $P$ in terms of $g$. [3]

CAIE Further Paper 3 2023 November Q5

9 marks Challenging +1.2

A particle $P$ is projected with speed $u\text{ ms}^{-1}$ at an angle $\theta$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. During its flight $P$ passes through the point which is a horizontal distance $3a$ from $O$ and a vertical distance $\frac{3}{8}a$ above the horizontal plane. It is given that $\tan\theta = \frac{1}{3}$.

Show that $u^2 = 8ag$. [2]

A particle $Q$ is projected with speed $V\text{ ms}^{-1}$ at an angle $\alpha$ above the horizontal from $O$ at the instant when $P$ is at its highest point. Particles $P$ and $Q$ both land at the same point on the horizontal plane at the same time.

Find $V$ in terms of $a$ and $g$. [7]

CAIE Further Paper 3 2023 November Q6

11 marks Challenging +1.8

A particle $P$ of mass $m$ is attached to one end of a light inextensible rod of length $3a$. An identical particle $Q$ is attached to the other end of the rod. The rod is smoothly pivoted at a point $O$ on the rod, where $OQ = x$. The system, of rod and particles, rotates about $O$ in a vertical plane. At an instant when the rod is vertical, with $P$ above $Q$, the particle $P$ is moving horizontally with speed $u$. When the rod has turned through an angle of $60°$ from the vertical, the speed of $P$ is $2\sqrt{ag}$, and the tensions in the two parts of the rod, $OP$ and $OQ$, have equal magnitudes.

Show that the speed of $Q$ when the rod has turned through an angle of $60°$ from the vertical is $\frac{2x}{3a-x}\sqrt{ag}$. [2]
Find $x$ in terms of $a$. [5]
Find $u$ in terms of $a$ and $g$. [4]

CAIE Further Paper 3 2023 November Q1

4 marks Standard +0.3

One end of a light inextensible string of length $a$ is attached to a fixed point $O$. The other end of the string is attached to a particle of mass $m$. The string is taut and makes an angle $\theta$ with the downward vertical through $O$, where $\cos \theta = \frac{2}{3}$. The particle moves in a horizontal circle with speed $v$. Find $v$ in terms of $a$ and $g$. [4]

CAIE Further Paper 3 2023 November Q2

6 marks Challenging +1.2

A particle $P$ of mass $0.5$ kg moves in a straight line. At time $t$ s the velocity of $P$ is $v$ m s$^{-1}$ and its displacement from a fixed point $O$ on the line is $x$ m. The only forces acting on $P$ are a force of magnitude $\frac{150}{(x+1)^2}$ N in the direction of increasing displacement and a resistive force of magnitude $\frac{450}{(x+1)^3}$ N. When $t = 0$, $x = 0$ and $v = 20$. Find $v$ in terms of $x$, giving your answer in the form $v = \frac{Ax + B}{(x + 1)}$, where $A$ and $B$ are constants to be determined. [6]

CAIE Further Paper 3 2023 November Q3

7 marks Challenging +1.2

\includegraphics{figure_3} A uniform lamina is in the form of an isosceles triangle $ABC$ in which $AC = 2a$ and angle $ABC = 90°$. The point $D$ on $AB$ is such that the ratio $DB : AB = 1 : k$. The point $E$ on $CB$ is such that $DE$ is parallel to $AC$. The triangle $DBE$ is removed from the lamina (see diagram).

Find, in terms of $k$, the distance of the centre of mass of the remaining lamina $ADEC$ from the midpoint of $AC$. [4]

When the lamina $ADEC$ is freely suspended from the vertex $A$, the edge $AC$ makes an angle $\theta$ with the downward vertical, where $\tan \theta = \frac{2}{15}$.

Find the value of $k$. [3]

CAIE Further Paper 3 2023 November Q4

7 marks Challenging +1.8

\includegraphics{figure_4} Two smooth vertical walls meet at right angles. The smooth sphere $A$, with mass $m$, is at rest on a smooth horizontal surface and is at a distance $d$ from each wall. An identical smooth sphere $B$ is moving on the horizontal surface with speed $u$ at an angle $\theta$ with the line of centres when the spheres collide (see diagram). After the collision, the spheres take the same time to reach a wall. The coefficient of restitution between the spheres is $\frac{1}{2}$.

Find the value of $\tan \theta$. [4]
Find the percentage loss in the total kinetic energy of the spheres as a result of this collision. [3]

CAIE Further Paper 3 2023 November Q5

8 marks Challenging +1.8

\includegraphics{figure_5} A bead of mass $m$ moves on a smooth circular wire, with centre $O$ and radius $a$, in a vertical plane. The bead has speed $v_A$ when it is at the point $A$ where $OA$ makes an angle $\alpha$ with the downward vertical through $O$, and $\cos \alpha = \frac{2}{3}$. Subsequently the bead has speed $v_B$ at the point $B$, where $OB$ makes an angle $\theta$ with the upward vertical through $O$. Angle $AOB$ is a right angle (see diagram). The reaction of the wire on the bead at $B$ is in the direction $OB$ and has magnitude equal to $\frac{1}{6}$ of the magnitude of the reaction when the bead is at $A$.

Find, in terms of $m$ and $g$, the magnitude of the reaction at $B$. [6]
Given that $v_A = \sqrt{kag}$, find the value of $k$. [2]

CAIE Further Paper 3 2023 November Q6

9 marks Standard +0.8

A particle $P$ is projected with speed $u$ at an angle $\alpha$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of $P$ from $O$ at a subsequent time $t$ are denoted by $x$ and $y$ respectively.

Derive the equation of the trajectory of $P$ in the form $$y = x \tan \alpha - \frac{gx^2}{2u^2} \sec^2 \alpha.$$ [3]

During its flight, $P$ must clear an obstacle of height $h$ m that is at a horizontal distance of $32$ m from the point of projection. When $u = 40\sqrt{2}$ m s$^{-1}$, $P$ just clears the obstacle. When $u = 40$ m s$^{-1}$, $P$ only achieves $80\%$ of the height required to clear the obstacle.

Find the two possible values of $h$. [6]

CAIE Further Paper 3 2023 November Q7

9 marks Challenging +1.8

\includegraphics{figure_7} A particle $P$ of mass $m$ is attached to one end of a light rod of length $3a$. 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 $kmg$. 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 $5a$ 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 $4a$ (see diagram). The particle is released from rest in this position and has a speed of $\frac{6}{5}\sqrt{2ag}$ when the rod becomes horizontal.

Find the value of $k$. [5]
Find $F$ in terms of $m$ and $g$. [2]
Find, in terms of $m$ and $g$, the tension in the rod immediately before it is released. [2]