Further Mechanics (for Paper 3): past paper questions

Where the questions come from

• Question 1Oct/Nov 2024, paper 31Structured5 marks

  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.

  Try this question
• Question 2May/June 2026, paper 31Structured5 marksHas a diagram

  The point $B$ is vertically below the point $A$. One end of a light inextensible string is attached to $A$, and the other end is attached to $B$. A particle $P$ of mass $7\text{ kg}$ is attached to the string so that the length $AP$ is $15\text{ m}$.

  Try this question
• Question 2Oct/Nov 2025, paper 31Structured4 marks

  A particle $P$ of mass $m$ is moving in a horizontal circle with angular speed ${\omega }_1$ on the smooth inner surface of a hemispherical shell of radius $r$.

  Try this question
• Question 2Oct/Nov 2024, paper 31Structured5 marks

  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.

  Try this question
• Question 2Oct/Nov 2023, paper 31Structured7 marks

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

  Try this question
• Question 3May/June 2026, paper 31Structured9 marks

  A particle $P$ of mass $6.4\text{ kg}$ moves along a straight horizontal line. At time $t\text{ s}$, the speed of $P$ is $v{\text{ ms}}^{-1}$ and the distance of $P$ from its starting position is $x\text{ m}$.

  Try this question
• Question 3Oct/Nov 2025, paper 31Structured7 marks

  A particle $P$ is moving in a straight horizontal line. At time $t\mathrm{s}$, the displacement of $P$ from a fixed point $O$ on the line is $x\mathrm{m}$ and the velocity of $P$ is $v{\mathrm{m}\mathrm{s}}^{-1}$.

  Try this question
• Question 3Oct/Nov 2024, paper 31Structured8 marks

  A particle $P$ of mass $m\mathrm{k}\mathrm{g}$ is attached to one end of a light elastic string of natural length $2\mathrm{m}$ and modulus of elasticity $2mg\mathrm{N}$. The other end of the string is attached to a fixed point $O$.

  Try this question
• Question 4May/June 2026, paper 31Structured8 marks

  A particle $P$ is moving on a fixed smooth horizontal surface with speed $u$. The particle collides with a fixed smooth vertical barrier. Immediately before the collision, the direction of motion of $P$ makes an angle $\alpha$ with the barrier, where $\tan \alpha =\frac{12}{5}$.

  Try this question
• Question 4Oct/Nov 2024, paper 31Structured7 marksHas a diagram

  An object is formed by removing a cylinder of radius $\frac{2}{3}a$ and height $kh$ ($k<1$) from a uniform solid cylinder of radius $a$ and height $h$. The vertical axes of symmetry of the two cylinders coincide.

  Try this question
• Question 5May/June 2026, paper 31Structured9 marks

  A particle of mass $m$ is attached to a fixed point $O$ by a light inextensible string of length $a$. The particle initially hangs directly below $O$ at the point $A$. The particle is then projected horizontally with speed $2\sqrt{ag}$, and begins to move in a vertical circle.

  Try this question
• Question 5Oct/Nov 2025, paper 31Structured9 marks

  A uniform lamina $OABCD$ consists of a rectangle $OACD$ and a triangle $ABC$. The length of $OA$ is $ka$, the length of $OD$ is $2a$, the height of triangle $ABC$ is $h$ and angle $CAB$ is ${45}^{\circ }$ (see diagram).

  Try this question
• Question 5Oct/Nov 2024, paper 31Structured7 marks

  A particle $P$ of mass $2\mathrm{k}\mathrm{g}$ 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 $t\mathrm{s}$.

  Try this question
• Question 5Oct/Nov 2023, paper 31Structured9 marks

  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.

  Try this question
• Question 6May/June 2026, paper 31Structured6 marks

  A light elastic string has natural length $2.5\text{ m}$ and modulus of elasticity $15\text{ N}$. One end of the string is attached to the fixed point $O$, and the other end is attached to a particle $P$ of mass $1.5\text{ kg}$.

  Try this question
• Question 6Oct/Nov 2025, paper 31Structured8 marksHas a diagram

  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 $O$. Initially $P$ is held with the string taut and making an angle of ${60}^{\circ }$ with the upward vertical through $O$.

  Try this question
• Question 6Oct/Nov 2024, paper 31Structured8 marksHas a diagram

  A particle $P$ of mass $0.05\mathrm{k}\mathrm{g}$ is attached to one end of a light inextensible string of length $1\mathrm{m}$. The other end of the string is attached to a fixed point $O$.

  Try this question
• Question 7May/June 2026, paper 31Structured6 marks

  The points $O$ and $A$ lie on a horizontal plane $10\text{ m}$ apart. The point $A$ lies at the foot of another plane which is inclined at an angle of $\alpha$ above the horizontal such that $\tan \alpha =\frac{4}{3}$.

  Try this question
• Question 7Oct/Nov 2025, paper 31Structured12 marks

  A particle $P$ is projected from a point $O$ on a horizontal plane and moves freely under gravity. The initial velocity of $P$ is $25{\mathrm{m}\mathrm{s}}^{-1}$ at an angle $\theta$ above the horizontal, where $\tan \theta =\frac{4}{3}$.

  Try this question
• Question 7Oct/Nov 2024, paper 31Structured10 marks

  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 $A$.

  Try this question
• Question 1Oct/Nov 2025, paper 32Structured4 marksHas a diagram

  A particle $P$ of mass $m$ is attached to two light inextensible strings each of length $l$. The end of one string is attached to a fixed point $A$ and the end of the other string is attached to a fixed point $B$, with $A$ vertically above $B$. Angle $APB$ is a right angle.

  Try this question
• Question 1Oct/Nov 2024, paper 32Structured3 marks

  A particle of mass $2\text{ kg}$ is attached to one end of a light elastic string of natural length $0.8\text{ m}$ and modulus of elasticity $100\text{ N}$. The other end of the string is attached to a fixed point $O$ on a smooth horizontal surface.

  Try this question
• Question 3Oct/Nov 2025, paper 32Structured6 marksHas a diagram

  A uniform lamina $OABCD$ is in the form of a rectangle, $OBCD$, joined along the edge $OB$ to a quarter circle $OAB$. The length of $DO$ is $ka$ and the length of $OB$ is $a$. The lamina rests in a vertical plane with its edge $CB$ on a horizontal surface (see diagram).

  Try this question
• Question 3Oct/Nov 2024, paper 32Structured7 marksHas a diagram

  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 $2u$ and $3u$ respectively.

  Try this question
• Question 4Oct/Nov 2025, paper 32Structured7 marks

  A particle $Q$ is initially positioned at a distance $d$ vertically above a particle $P$. Particle $P$ is projected with speed $U$ at an angle $\alpha$ above the horizontal. At the same time, $Q$ is projected at an angle $\beta$ below the horizontal.

  Try this question
• Question 5Oct/Nov 2025, paper 32Structured9 marksHas a diagram

  Two uniform smooth spheres, $A$ and $B$, of equal radii are on a horizontal surface. They have masses of $m$ and $km$ respectively. Sphere $A$ is moving with speed $u$ at an angle $\alpha$ with the line of centres when it collides with sphere $B$ which is stationary.

  Try this question
• Question 5Oct/Nov 2024, paper 32Structured8 marks

  A particle $P$ is projected from a point $O$ on horizontal ground with speed $u$ at an angle $\theta$ above the horizontal, where $\tan \theta =\frac{1}{3}$.

  Try this question
• Question 6Oct/Nov 2025, paper 32Structured8 marksHas a diagram

  A fixed smooth sphere with radius $a$ and centre $O$ rests on horizontal ground. A particle is projected horizontally from the highest point, $A$, of the sphere with speed $u$. The particle begins to move in a vertical circle along the surface of the sphere.

  Try this question
• Question 7Oct/Nov 2025, paper 32Structured10 marks

  A particle $P$ of mass $m\text{ kg}$ moving along a rough horizontal table has displacement $x\text{ m}$ from a fixed point $O$ on the table and velocity $v{\text{ ms}}^{-1}$ at time $t\text{ s}$.

  Try this question
• Question 7Oct/Nov 2024, paper 32Structured11 marks

  A particle $P$ of mass $m\text{ kg}$ is held at rest at a point $O$ and released so that it moves vertically under gravity against a resistive force of magnitude $0.1m{v}^2\text{ N}$, where $v{\text{ ms}}^{-1}$ is the velocity of $P$ at time $t\text{ s}$.

  Try this question