Kinematics of motion in a straight line

A particle $P$ travels along a straight line. Its velocity $v\,\text{m s}^{-1}$ at time $t\,\text{s}$ is defined by $v = 5t(t - 2)$ for $0 \leq t \leq 4$, $v = k$ for $4 \leq t \leq 14$, $v = 68 - 2t$ for $14 \leq t \leq 20$, where $k$ is constant.

Feb/March 2016

A car with mass $900\text{ kg}$ travels along the straight horizontal road $ABCD$. On $AB$ and $BC$, the resistive force has constant magnitude $800\text{ N}$, while on $CD$ it has constant magnitude $R\text{ N}$. The engine supplies a constant power of $36\text{ kW}$.

Feb/March 2017

A particle $P$ travels in a straight line from a point $O$ and is brought to rest $35\,\text{s}$ later. For $t\,\text{s}$ after leaving $O$, the velocity $v\,\text{m s}^{-1}$ of $P$ is given by $v = \frac{4}{5}t^2 \quad 0 \leq t \leq 5,$ $v = 2t + 10 \quad 5 \leq t \leq 15,$ $v = a + bt^2 \quad 15 \leq t \leq 35,$ where $a$ and $b$ are constants with $a > 0$ and $b < 0$.

Feb/March 2017

A small rocket is launched straight up from ground level from rest, and it travels with constant acceleration. After $10\,\text{s}$, the rocket is at a height of $200\,\text{m}$.

Feb/March 2018

A particle $P$ travels along a straight line. Its velocity $v\,\text{m s}^{-1}$ at time $t\,\text{s}$ is defined by $v = 4 + 0.2t$ for $0 \le t \le 10$, and by $v = -2 + \frac{800}{t^2}$ for $10 \le t \le 20$.

Feb/March 2018

A particle is launched vertically upwards at a speed of $30\,\text{m s}^{-1}$ from a point on horizontal ground.

Feb/March 2019

For a particle travelling along a straight line, its velocity is $v\,\text{m s}^{-1}$ at $t$ seconds after it leaves the fixed point $O$. The diagram displays a velocity-time graph that represents the particle's motion for $t=0$ to $t=16$. It is made up of five straight line sections. The acceleration from $t=0$ to $t=3$ is $3\,\text{m s}^{-2}$. At $t=5$, the particle's velocity is $7\,\text{m s}^{-1}$ and it is momentarily at rest at $t=8$. It then is at rest again at $t=16$. The smallest velocity reached is $V\,\text{m s}^{-1}$.

Feb/March 2019

A particle travels in a straight line. It begins from rest at a fixed point $O$ on the line. At time $t$ seconds after leaving $O$, its acceleration is $a\,\text{m s}^{-2}$, where $a=0.4t^{3}-4.8t^{\frac{1}{2}}$.

Feb/March 2019

A particle $P$ with mass $0.4\,\text{kg}$ rests on a rough horizontal floor. The coefficient of friction between $P$ and the floor is $\mu$. A force of magnitude $3\,\text{N}$ acts on $P$ upwards at an angle $\alpha$ above the horizontal, where $\tan \alpha = \frac{3}{4}$. Initially the particle is at rest, and its acceleration is $2\,\text{m}\,\text{s}^{-2}$.

Feb/March 2020

A cyclist moves on a straight road with constant acceleration. He goes through points $A$, $B$ and $C$. He takes $2$ seconds to cover each of the segments $AB$ and $BC$, and his speed at $B$ is $4.5\,\text{m s}^{-1}$. The length of $AB$ is $\frac{4}{5}$ of the length of $BC$.

Feb/March 2020

A particle travels along a straight line through point $O$. Its displacement from $O$ at time $t$ is $s\,\text{m}$, where $s = t^2 - 3t + 2$ for $0 \le t \le 6$, and $s = \frac{24}{t} - \frac{t^2}{4} + 25$ for $t > 6$.

Feb/March 2020

A particle travels along a straight line. It begins from rest at a fixed point $O$ on the line. At time $t\text{ s}$ after leaving $O$, its velocity is $v\text{ m s}^{-1}$, where $v = t^2 - 8t^{\frac{3}{2}} + 10t$.

Feb/March 2021

A particle $P$ is launched vertically upwards from level ground with speed $u\\,\\text{m}\\,\\text{s}^{-1}$. $P$ attains a greatest height of $20\\,\\text{m}$ above the ground.

Feb/March 2022

A cyclist sets off from rest at the fixed point $O$ and travels in a straight line, then comes to rest $k$ seconds later. At time $t$ s after leaving $O$, the cyclist’s acceleration is $a\,\text{m s}^{-2}$, where $a = 2t^{-\frac{1}{2}} - \frac{3}{5}t^{\frac{1}{2}}$ for $0 < t \le k$.

Feb/March 2022

A crate with mass $200\,\text{kg}$ is being dragged at constant speed across horizontal ground by a horizontal rope connected to a winch. The winch operates at a steady rate of $4.5\,\text{kW}$, and the crate experiences a constant resistive force with magnitude $600\,\text{N}$.

Feb/March 2023

A particle $P$ is projected straight upwards from horizontal ground with initial speed $15\,\text{m s}^{-1}$.

Feb/March 2023

A particle travels in a straight line, beginning from rest at point $O$. At time $t\,\text{s}$ after it leaves $O$, its acceleration is $a\,\text{m s}^{-2}$, where $a = 4t^{\frac{1}{2}}$.

Feb/March 2023

The displacement of a particle after it has left a fixed point $O$ at time $t$ is $s$ m. The diagram presents a displacement-time graph that represents the particle's motion. The graph is made up of $4$ straight line segments. In the first $10$ s, the particle covers $50$ m, then it moves at $2\,\text{m s}^{-1}$ for $10$ s. After that, the particle remains at rest for $20$ s, before it gets back to its starting point when $t = 60$.

Feb/March 2024

A particle is launched vertically upwards from horizontal ground. Two seconds after launch, its speed is $5\,\text{m s}^{-1}$ and it is moving downwards.

Feb/March 2024

A particle travels along a straight line from a point $O$. Its velocity $v\text{ m s}^{-1}$, $t\text{ s}$ after it leaves $O$, is $v = t^3 - \frac{9}{2}t^2 + 1$ for $0 \leq t \leq 4$. You may take the velocity to be positive when $t < \frac{1}{2}$, to be zero when $t = \frac{1}{2}$ and to be negative when $t > \frac{1}{2}$.

Feb/March 2024

A cyclist moves along a straight horizontal road at a speed of $4\,\text{m s}^{-1}$ as she passes point $O$. She then accelerates uniformly over a distance of $42\,\text{m}$ until her speed becomes $V\,\text{m s}^{-1}$. After that, she continues at $V\,\text{m s}^{-1}$ for $50\,\text{m}$ and then slows down at $2\,\text{m s}^{-2}$ before stopping completely. The distance covered during deceleration is $16\,\text{m}$.

Feb/March 2025

A particle travels along a straight line. Its velocity $v\,\text{m s}^{-1}$, $t\,\text{s}$ after it leaves a fixed point $O$, is defined by $v = k(20 + pt - 6t^2)$, where $k$ and $p$ are constants. At $t = 1$, the particle has acceleration $42\,\text{m s}^{-2}$ and displacement from $O$ equal to $93\,\text{m}$.

Feb/March 2025

The diagram presents the velocity-time graph for a machine cutting tool’s motion. It is made up of five straight-line sections. The tool travels forward for $8 \text{ s}$ while cutting, then needs $3 \text{ s}$ to get back to its starting point.

May/June 2010

A vehicle travels along a straight line. Its velocity $v \text{ m s}^{-1}$ at time $t \text{ s}$ after it begins is given by $$v = A(t - 0.05t^{2}) \text{ for } 0 \le t \le 15,$$ $$v = \frac{B}{t^{2}} \text{ for } t > 15,$$ where $A$ and $B$ are constants. The distance the vehicle covers from $t = 0$ to $t = 15$ is $225 \text{ m}$.

May/June 2010

The diagram presents a velocity-time graph for the motion of a machine’s cutting tool. It is made up of five straight-line sections. The tool travels forwards for $8\,\text{s}$ while it is cutting and then needs $3\,\text{s}$ to get back to its starting point.

May/June 2010

A vehicle travels along a straight line. Its velocity $v\,\text{m s}^{-1}$, measured at time $t\,\text{s}$ after the start of motion, is defined by $v = A(t - 0.05t^2)$ for $0 \le t \le 15$, and by $v = \frac{B}{t^2}$ for $t > 15$, where $A$ and $B$ are constants. The distance covered by the vehicle from $t = 0$ to $t = 15$ is $225\,\text{m}$.

May/June 2010

A particle is set off from point $O$ and travels along a straight line. Its velocity $t\,\text{s}$ after leaving $O$ is $(1.2t - 0.12t^2)\,\text{m s}^{-1}$. Find the displacement of the particle from $O$ when its acceleration is $0.6\,\text{m s}^{-2}$.

May/June 2010

A ball travels along the horizontal surface of a billiards table, slowing down with constant deceleration of magnitude $d\,\text{m s}^{-2}$. It leaves $A$ at speed $1.4\,\text{m s}^{-1}$ and, $1.2\,\text{s}$ later, arrives at the table edge at $B$ with speed $1.1\,\text{m s}^{-1}$.

May/June 2010

A train leaves station $A$ at rest and moves along a straight track to station $B$, where it finally stops. For the first $600\,\text{s}$ it has constant acceleration $0.025\,\text{m s}^{-2}$, then it continues at constant speed for the next $2600\,\text{s}$, and at the end it slows down with constant deceleration $0.0375\,\text{m s}^{-2}$.

May/June 2011

A particle moves along a straight line from point $P$ to point $Q$. $t$ seconds after leaving $P$, its velocity is $v\,\text{m s}^{-1}$, where $v = 4t - \frac{1}{16}t^3$. The distance $PQ$ is $64\,\text{m}$.

May/June 2011

Particles $P$ and $Q$ are launched vertically upwards from level ground at the same moment. Their initial speeds are $12\,\text{m s}^{-1}$ and $7\,\text{m s}^{-1}$ respectively, and $t$ seconds after launch their heights above the ground are $h_P$ and $h_Q$ respectively. On the way back down to the ground, each particle comes to rest.

May/June 2011

A walker moves along a straight road through points $A$ and $B$, with speeds $0.9\,\text{m s}^{-1}$ and $1.3\,\text{m s}^{-1}$ at $A$ and $B$ respectively. The acceleration from $A$ to $B$ is constant, $0.004\,\text{m s}^{-2}$. A cyclist departs from $A$ at the same moment as the walker. She begins from rest and rides along the straight road, reaching $B$ at the same instant as the walker. After time $t$ seconds from leaving $A$, the cyclist’s speed is $kt^2\,\text{m s}^{-1}$, where $k$ is a constant.

May/June 2011

The velocity-time graphs in the diagram describe the motion of two particles $P$ and $Q$, both moving in the same direction along a straight line. $P$ and $Q$ begin from the same point $X$ on the line, although $Q$ begins $T\,\text{s}$ after $P$. For the first $20\,\text{s}$ of motion, each particle travels at speed $2.5\,\text{m s}^{-1}$. Once each has been moving for $20\,\text{s}$, its speed increases instantaneously to $4\,\text{m s}^{-1}$ and then stays at that speed. It is given that $P$ has covered $70\,\text{m}$ at the moment $Q$ begins.

May/June 2011

A particle moves along a straight line from $A$ to $B$ in $20\,\text{s}$. At $t$ seconds after departing from $A$, its acceleration is $a\,\text{m s}^{-2}$, where $a = \frac{3}{160}t^2 - \frac{1}{800}t^3$. It is stated that the particle is at rest when it reaches $B$.

May/June 2011

A particle $P$ begins at $O$ and moves along a straight line. After $t$ seconds from leaving $O$, its velocity is $v\,\text{m s}^{-1}$, where $v = 0.75t^2 - 0.0625t^3$.

May/June 2012

A particle $P$ travels along a straight line, beginning at $O$ with velocity $2\,\text{m s}^{-1}$. The acceleration of $P$ at time $t\,\text{s}$ after leaving $O$ is $2t^{\frac{2}{3}}\,\text{m s}^{-2}$.

May/June 2012

A frictional force of magnitude $0.12\,\text{N}$ acts on a small block of mass $0.15\,\text{kg}$ while it is travelling over a horizontal surface. The block is launched from a point $X$ on the surface with speed $3\,\text{m s}^{-1}$. After $2\,\text{s}$ it reaches a vertical wall at point $Y$ on the surface. It rebounds from the wall and then travels straight back towards $X$ until it stops at point $Z$ (see diagram). When the block strikes the wall, its kinetic energy decreases by $0.072\,\text{J}$. At time $t\,\text{s}$ after leaving $X$, the velocity of the block in the direction from $X$ to $Y$ is $v\,\text{m s}^{-1}$.

May/June 2012

A particle $P$ moves from point $O$ in a straight line and reaches instantaneous rest at point $A$. The velocity of $P$ after time $t$ since leaving $O$ is $v\,\text{m s}^{-1}$, where $v = 0.027(10t^2 - t^3)$.

May/June 2012

Particles $A$ and $B$ have masses $0.12\,\text{kg}$ and $0.38\,\text{kg}$ respectively. They are connected by the ends of a light inextensible string, which goes over a fixed smooth pulley. $A$ is kept stationary with the string taut, and both sections of the string are vertical. Each particle is $0.65\,\text{m}$ above the horizontal ground (see diagram). $A$ is then released, and $B$ moves downwards.

May/June 2012

A cliff top is $40\text{ metres}$ above sea level. A man in a boat, near the foot of the cliff, is in trouble and launches a distress signal vertically upward from sea level.

May/June 2013

A car driver travels in a straight line from $A$ to $B$, beginning from rest. The car’s speed rises to a highest value and then falls until it is again at rest at $B$. $t$ seconds after leaving $A$, the distance covered by the car is $0.0000117(400t^3 - 3t^4)\text{ metres}$.

May/June 2013

A particle $P$ is let go from rest at the top of a smooth plane inclined at an angle $\alpha$ to the horizontal, where $\sin\alpha = \frac{16}{65}$. The distance covered by $P$ from the top to the bottom is $S\,\text{m}$, and the speed of $P$ at the bottom is $8\,\text{m s}^{-1}$. The time taken by $P$ to move from the top to the bottom of the plane is $T\,\text{s}$.

May/June 2013

A particle $P$ travels along a straight line. It begins from rest at point $O$ and heads towards point A on the line. For the first $8\,\text{s}$, its speed rises to $8\,\text{m s}^{-1}$ with constant acceleration. Over the following $12\,\text{s}$, its speed falls to $2\,\text{m s}^{-1}$ with constant deceleration. $P$ then continues with constant acceleration for $6\,\text{s}$, arriving at A with speed $6.5\,\text{m s}^{-1}$. The displacement of $P$ from $O$, at time $t$ seconds after $P$ leaves $O$, is $s$ metres.

May/June 2013

An aeroplane travels in a straight horizontal line along the runway before lifting off. It is at rest at $O$ and has speed $90\,\text{m s}^{-1}$ at the moment it takes off. At time $t$ seconds after leaving $O$, while it is still on the runway, its acceleration is $(1.5 + 0.012t)\,\text{m s}^{-2}$.

May/June 2013

A particle $P$ is launched vertically upwards from a point on the ground at a speed of $17\,\text{m s}^{-1}$. A second particle $Q$ is launched vertically upwards from the same point at a speed of $7\,\text{m s}^{-1}$. Particle $Q$ is launched $T$ seconds after particle $P$.

May/June 2013

A train is travelling at a constant speed $V\,\text{m s}^{-1}$ on a horizontal straight track. The engine power is $1330\,\text{kW}$, and the total resistance opposing the train’s motion is $28\,\text{kN}$.

May/June 2014

A particle is launched vertically upward with speed $9\,\text{m s}^{-1}$ from a point $3.15\,\text{m}$ above horizontal ground. It then travels freely under gravity until it reaches the ground.

May/June 2014

Cyclists $P$ and $Q$ move along the straight road $ABC$, setting off together from $A$ and reaching $C$ at the same time. They each pass through $B$ $400\,\text{s}$ after leaving $A$. Cyclist $P$ begins with speed $3\,\text{m s}^{-1}$ and then increases this speed with constant acceleration $0.005\,\text{m s}^{-2}$ until he arrives at $B$.

May/June 2014

Particle $P$ travels along a straight line, beginning from rest at point $O$ on the line. Let $t$ s denote the time since $P$ began moving. The particle then moves along the line with constant acceleration $\frac{1}{4}\,\text{m s}^{-2}$ until it goes through point $A$ when $t = 8$. Once it has passed through $A$, the velocity of $P$ is $\frac{1}{2}t^{3/4}\,\text{m s}^{-1}.

May/June 2014

A small ball with mass $0.4\text{ kg}$ is let go from rest from a point $5\text{ m}$ above the horizontal ground. The moment the ball reaches the ground, $12.8\text{ J}$ of kinetic energy is lost and it then begins to rise.

May/June 2014

A particle is released from rest at point $O$ and travels along a horizontal straight line. At time $t\text{ s}$ after leaving $O$, its velocity is $v\text{ m s}^{-1}$. When $0 \le t < 60$, the velocity is given by $v = 0.05t - 0.0005t^2$. At $t = 60$, the particle strikes a wall and then changes the direction of its motion. It later comes to rest at point $A$ when $t = 100$, and for $60 < t \le 100$ the velocity is given by $v = 0.025t - 2.5$.

May/June 2014

Particles $A$ and $B$ start moving at the same moment from point $O$. They travel in the same direction along one straight line. At time $t\,\text{s}$ after the motion begins, the acceleration of $A$ is $a\,\text{m s}^{-2}$, where $a = 0.05 - 0.0002t$.

May/June 2015

A particle $P$ travels along a straight line. $t$ seconds after it starts from rest at the point $O$ on the line, its acceleration is $a\,\text{m s}^{-2}$, where $a = 0.075t^{2} - 1.5t + 5$.

May/June 2015

Particle $P$ begins from rest at point $O$ on a horizontal straight line. $P$ travels along the line with constant acceleration and arrives at point $A$ on the line with a speed of $30\,\text{m s}^{-1}$. At the moment that $P$ departs from $O$, particle $Q$ is projected vertically upwards from point $A$ with a speed of $20\,\text{m s}^{-1}$. Later, $P$ and $Q$ collide at $A$. Find

May/June 2015

A particle $P$ travels along a straight line. It begins at a point $O$ on the line and is back at $O\,100\,\text{s}$ later. The velocity of $P$ is $v\,\text{m s}^{-1}$ at time $t\,\text{s}$ after leaving $O$, where $v = 0.0001t^{3} - 0.015t^{2} + 0.5t$.

May/June 2015

A lift starts from rest and increases its speed upwards at $0.9\,\text{m s}^{-2}$ for $3\,\text{s}$. It then continues for $6\,\text{s}$ at a steady speed and, after that, slows with a constant deceleration until it stops in another $4\,\text{s}$.

May/June 2016

Particle $P$ travels along a straight line. It begins at point $O$ on the line and, $t\,\text{s}$ after leaving $O$, its velocity is $v\,\text{m s}^{-1}$, where $v = 6t^2 - 30t + 24$.

May/June 2016

Particle $P$ travels along a straight line, beginning at point $O$. For time $t$ after it has left $O$, the velocity of $P$, in $\text{m s}^{-1}$, is $v = 4t^2 - 8t + 3$.

May/June 2016

A sprinter takes part in a $400\,\text{m}$ race, and his total running time is $52\,\text{s}$. The diagram gives the velocity-time graph for the sprinter’s motion. He begins from rest and increases his speed uniformly to $8.2\,\text{m s}^{-1}$ in $6\,\text{s}$. He then keeps a speed of $8.2\,\text{m s}^{-1}$ for $36\,\text{s}$ before slowing down uniformly to a speed of $V\,\text{m s}^{-1}$ at the finish.

May/June 2016

Alan begins moving from point $O$ at a steady speed of $4\,\text{m s}^{-1}$, travelling along a horizontal path. Ben travels on the same path and also sets off from $O$. Ben remains at rest for $5\,\text{s}$ after Alan begins, then accelerates at $1.2\,\text{m s}^{-2}$ for $5\,\text{s}$. After that, Ben keeps moving at a constant speed until he reaches the same point, $P$, as Alan.

May/June 2016

A particle $P$ travels along a straight line. At time $t\,\text{s}$, its displacement from $O$ is $s\,\text{m}$, and its acceleration is $a\,\text{m s}^{-2}$, where $a = 6t - 2$. When $t = 1$, $s = 7$ and when $t = 3$, $s = 29$.

May/June 2016

A particle with mass $0.8\text{ kg}$ is launched at a speed of $12\text{ m s}^{-1}$ up the line of greatest slope on a rough plane inclined at $10^\circ$ to the horizontal. The coefficient of friction between the particle and the plane is $0.4$.

May/June 2017

A particle $P$ travels along the straight line $ABCD$ with constant deceleration. Its velocities at $A$, $B$ and $C$ are $20\,\text{m s}^{-1}$, $12\,\text{m s}^{-1}$ and $6\,\text{m s}^{-1}$ respectively.

May/June 2017

A particle $P$ travels along a straight line through a point $O$. When the time is $t\text{ s}$, the velocity of $P$, $v\text{ m s}^{-1}$, is given by $v = qt + rt^2$, where $q$ and $r$ are constants. The particle’s velocity is $4\text{ m s}^{-1}$ at both $t = 1$ and $t = 2$.

May/June 2017

Particle $A$ travels along a straight line at a constant speed of $10\,\text{m s}^{-1}$. Two seconds after $A$ has passed point $O$ on the line, particle $B$ passes through $O$ and moves along the line in the same direction as $A$. At $O$, particle $B$ has speed $16\,\text{m s}^{-1}$ and then undergoes a constant deceleration of $2\,\text{m s}^{-2}$.

May/June 2017

A train runs between stations $A$ and $B$. It leaves $A$ from rest and increases its speed at a constant rate for $T\,\text{s}$ until its speed becomes $25\,\text{m\,s}^{-1}$. It then continues at this same speed before slowing down at a constant rate and stopping at $B$. The size of the train’s deceleration is twice the size of its acceleration. The whole trip takes $180\,\text{s}$.

May/June 2017

A particle $P$ travels along a straight line from point $O$. After $t$ s has elapsed since it left $O$, its velocity, $v\,\text{m s}^{-1}$, is given by $v = (2t - 5)^3$.

May/June 2017

One particle is launched vertically upwards from point $O$ at a speed of $12\,\text{m s}^{-1}$. Two seconds afterwards, a second particle is launched vertically upwards from $O$ at a speed of $20\,\text{m s}^{-1}$. At time $t$ s after the second particle is launched, the two particles collide.

May/June 2017

A particle $P$ is launched vertically upwards at speed $24\,\text{m s}^{-1}$ from a position $5\,\text{m}$ above ground level.

May/June 2018

A particle $P$ travels along a straight line from $O$. After $t$ s from leaving $O$, its displacement from $O$ is $s$ m, where $s = t^3 - 4t^2 + 4t$, and its velocity is $v\,\text{m s}^{-1}$.

May/June 2018

A sprinter takes part in a race of $200\,\text{m}$. The total time taken to complete the race is $20\,\text{s}$. He begins from rest and speeds up uniformly for $6\,\text{s}$, reaching a speed of $12\,\text{m s}^{-1}$. He then keeps this speed for the following $10\,\text{s}$, before slowing down uniformly and crossing the finishing line with speed $V\,\text{m s}^{-1}$.

May/June 2018

A particle $P$ travels along the straight line $ABCD$ with uniform acceleration. The lengths $AB$ and $BC$ are $100\,\text{m}$ and $148\,\text{m}$, respectively. It takes the particle $4\,\text{s}$ to move from $A$ to $B$ and another $4\,\text{s}$ to move from $B$ to $C$.

May/June 2018

A particle $P$ travels along a straight line through a point $O$. At time $t\,\text{s}$, the acceleration, $a\,\text{m s}^{-2}$, of $P$ is $a = 6 - 0.24t$. The particle is instantaneously at rest at $t = 20$.

May/June 2018

The diagram illustrates the velocity-time graph for a train travelling from rest at one station to rest at the next. It is made up of three straight line segments. The distance separating the two stations is $9040\text{ m}$.

May/June 2018

A small ball is thrown straight downwards from point $A$, which is $7.2\text{ m}$ above level ground, with initial speed $5\text{ m s}^{-1}$. It strikes the ground at speed $V\text{ m s}^{-1}$ and then bounces straight up with speed $\frac{1}{2}V\text{ m s}^{-1}$. After the bounce, the ball rises to its greatest height at $B$.

May/June 2018

A particle $P$ travels along a straight line and begins at point $O$. Its velocity $v\text{ m s}^{-1}$ at time $t\text{ s}$ is defined by $v = 12t - 4t^2$ for $0 \le t \le 2$, and $v = 16 - 4t$ for $2 \le t \le 4$.

May/June 2018

Particle $P$ is fired vertically upwards from a point $3\,\text{m}$ above level ground, with initial speed $25\,\text{m s}^{-1}$.

May/June 2019

A particle $P$ travels along a straight line starting from a fixed point $O$. Its velocity $v\,\text{m s}^{-1}$ at time $t\,\text{s}$ is defined by $v = t^2 - 8t + 12$ for $0 \leq t \leq 8$.

May/June 2019

A car travels along a straight path with starting speed $u\,\mathrm{m\,s^{-1}}$ and uniform acceleration $a\,\mathrm{m\,s^{-2}}$. It needs $5\,\mathrm{s}$ to cover the initial $80\,\mathrm{m}$ and $8\,\mathrm{s}$ to cover the initial $160\,\mathrm{m}$.

May/June 2019

A force resisting motion with constant magnitude $350\,\text{N}$ acts on a car of mass $1250\,\text{kg}$. The car’s engine provides a constant driving force of $1200\,\text{N}$. The car moves along a road inclined at an angle of $\theta$ to the horizontal, where $\sin \theta = 0.05$.

May/June 2019

Particles $P$ and $Q$ depart from the fixed point $A$ at the same instant and move along the same straight line. After $t$ seconds, the velocity of $P$ is $6t(t - 3)\,\text{m s}^{-1}$, while the velocity of $Q$ is $(10 - 2t)\,\text{m s}^{-1}$.

May/June 2019

A bus travels in a straight line from one bus stop to another. It begins from rest and accelerates at $2.1\,\text{m s}^{-2}$ for $5\,\text{s}$. It then continues at a constant speed for $24\,\text{s}$ before slowing down with a constant deceleration and coming to rest after a further $6\,\text{s}$.

May/June 2019

A particle $P$ travels in a straight line. Its acceleration $a\,\text{m s}^{-2}$ at time $t\,\text{s}$ is $a = 6t - 12$. The displacement of $P$ from a fixed point $O$ on the line is $s\,\text{m}$. It is known that $s = 5$ when $t = 1$ and $s = 1$ when $t = 3$.

May/June 2019

A particle $P$ is launched vertically upwards at a speed of $5\,\text{m s}^{-1}$ from a point $A$, which lies $2.8\,\text{m}$ above the horizontal ground.

May/June 2020

A particle travels along the straight line $AB$. Its velocity $v\,\text{m s}^{-1}$ after $t\,\text{s}$ from leaving $A$ is $v = k(t^2 - 10t + 21)$, where $k$ is a constant. When $t = 3$, the particle's displacement from $A$, measured towards $B$, is $2.85\,\text{m}$, and when $t = 6$ it is $2.4\,\text{m}$.

May/June 2020

A tram begins at rest and accelerates uniformly for $20\text{ s}$. It then continues at a steady speed, $V\text{ m s}^{-1}$, for $170\text{ s}$ before coming to rest under a uniform deceleration whose magnitude is twice the acceleration. The tram covers a total distance of $2.775\text{ km}$.

May/June 2020

A particle $P$ moves along a straight line. The velocity $v\,\text{m s}^{-1}$ at time $t\,\text{s}$ is given by: $v = 2t + 1$ for $0 \leq t \leq 5$, $v = 36 - t^2$ for $5 \leq t \leq 7$, $v = 2t - 27$ for $7 \leq t \leq 13.5$.

May/June 2020

A car begins at rest and travels in a straight line with constant acceleration $a\,\text{m s}^{-2}$ over a distance of $50\,\text{m}$. It then moves at constant velocity for $500\,\text{m}$ in $25\,\text{s}$, before slowing to rest. The deceleration magnitude is $2a\,\text{m s}^{-2}$.

May/June 2020

A particle moves along the straight line $PQ$. Its velocity $t$ s after leaving $P$ is $v$ m s$^{-1}$, where $v = 4.5 + 4t - 0.5t^2$. The particle is instantaneously at rest at $Q$.

May/June 2020

Two cyclists, Isabella and Maria, are racing. Each one moves along a straight road with constant acceleration, beginning from rest at point $A$. Isabella accelerates for $5\,\text{s}$ at a steady rate of $a\,\text{m s}^{-2}$. After that, she continues at the constant speed she has reached for $10\,\text{s}$, and then she slows to rest at a constant rate over $5\,\text{s}$. Maria accelerates at a constant rate until she reaches $5\,\text{m s}^{-1}$ over a distance of $27.5\,\text{m}$. She then keeps this speed for $10\,\text{s}$ before decelerating to rest at a constant rate over $5\,\text{s}$.

May/June 2021

A particle travels in a straight line, beginning from rest at $A$ and stopping instantaneously at $B$. At time $t$ after it has left $A$, its acceleration is $a\,\text{m s}^{-2}$, where $a = 6t^2 - 2t$.

May/June 2021

A ring with mass $0.3\,\text{kg}$ is passed through a horizontal rough rod. The coefficient of friction between the ring and the rod is $0.8$. A force of magnitude $8\,\text{N}$ acts on the ring. It is applied at an angle of $10^\circ$ above the horizontal in the vertical plane containing the rod.

May/June 2021

Particle $A$ is launched vertically upward from level ground with an initial speed of $30\,\text{m s}^{-1}$. At that same instant, particle $B$ is let go from rest $15\,\text{m}$ directly above $A$. One of the particles has twice the mass of the other. As the motion proceeds, $A$ and $B$ collide and stick together to make particle $C$.

May/June 2021

Particle $P$ moves along a straight line, begins from rest at point $O$, and is back at rest $16\,\text{s}$ after leaving $O$. If $t\,\text{s}$ has elapsed since it left $O$, the acceleration $a\,\text{m s}^{-2}$ of $P$ is defined by $a = 6 + 4t\quad 0 \leq t < 2,$ $a = 14\quad 2 \leq t < 4,$ $a = 16 - 2t\quad 4 \leq t \leq 16.$ At no instant does the velocity change suddenly.

May/June 2021

A particle is launched vertically upwards at speed $u\,\text{m s}^{-1}$ from a point on level ground. After $2$ seconds, its height above the ground is $24\,\text{m}$.

May/June 2021

A particle travels along a straight line and passes through point $A$ at time $t = 0$. After leaving $A$, the particle’s velocity at time $t$ s is $v\,\text{m s}^{-1}$, where $v = 2t^2 - 5t + 3$.

May/June 2021

A car begins from rest and travels in a straight line with constant acceleration over a distance of $200\,\text{m}$, arriving at a speed of $25\,\text{m s}^{-1}$. It then continues at this same speed for $400\,\text{m}$, before slowing down uniformly to rest in $5\,\text{s}$.

May/June 2022

A particle leaves the point O and then travels along a straight line. The velocity v\text{ m s}^{-1} of the particle at time t\text{ s} after it has left O is given by v = k(3t^2 - 2t^3), where k is a constant.

May/June 2022

A particle $A$, travelling along a straight horizontal track at a constant speed of $8\text{ m s}^{-1}$, goes past a fixed point $O$. Four seconds later, a second particle $B$ goes past $O$, moving along a parallel track in the same direction as $A$. Particle $B$ is moving at speed $20\text{ m s}^{-1}$ as it passes $O$ and has a constant deceleration of $2\text{ m s}^{-2}$. $B$ comes to rest when it returns to $O$.

May/June 2022

Particle $P$ travels along a straight line. Its velocity $v\,\text{m s}^{-1}$ at time $t$ seconds is defined by $v = 0.5t$ for $0 \leq t \leq 10$, and by $v = 0.25t^2 - 8t + 60$ for $10 \leq t \leq 20$.

May/June 2022