Constant Acceleration in 2D (AQA A Level Maths: Mechanics): Exam Questions

A horse running across a large area of open countryside starts to gallop with constant acceleration $\left(0.25\mathbf{i}+0.45\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-2}$.

After 16 seconds of galloping the horse has velocity $\left(12\mathbf{i}+6\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-1}$.

Find the displacement of the horse after 16 seconds, relative to the point where it started to gallop.

Hence find the average velocity of the horse during the gallop.

[In this question, $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin $O$.]

A particle $P$ moves with constant acceleration $(2\mathbf{i}-3\mathbf{j}){\mathrm{ms}}^{-2}$. At time $t=0$, the particle is at the point $A$ with position vector $(4\mathbf{i}+\mathbf{j})\mathrm{m}$ and is moving with velocity $(-\mathbf{i}+4\mathbf{j}){\mathrm{ms}}^{-1}$.

Find the speed of $P$ at time $t=3$ seconds.

At time $t=T$ seconds, where $T>0$, the particle passes through the point $B$.

Given that the position vector of $B$ is $(\lambda \mathbf{i}-7\mathbf{j})\mathrm{m}$, where $\lambda$ is a constant, find the value of $T$.

It takes four minutes for a particle to travel $\left(\begin{array}{c}0.96\\ 1.2\end{array}\right)$ km with constant acceleration. 

The velocity of the particle at the end of the four minutes is triple the velocity of the particle at the start. 

Find the initial and final velocities, giving your answers in metres per second.

A football is kicked from the top of a hill and its motion is modelled as that of a particle moving in a 2D vertical plane with constant acceleration.

The initial velocity of the football is $\left(18\mathbf{i}+23\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-1}$ and it lands at ground level with velocity $\left(18\mathbf{i}-26\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-1}$, where i is a unit vector in the horizontal direction and j is a unit vector in the upwards vertical direction.

Find the time it takes the football to first hit ground level.

Find the displacement of the football when it first hits ground level, relative to the point where it was kicked.

Starting from rest, a toy boat experiences a constant acceleration of $\left(0.5\mathbf{i}+0.2\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-2}$.

The toy boat takes 12 seconds to sail across a pond.  

Find the distance that the toy boat sails across the pond, giving your answer to three significant figures.

A particle with initial velocity $\left(\begin{array}{c}2q-1\\ q\end{array}\right){\mathrm{ms}}^{-1}$ accelerates for 9 seconds to reach a velocity of $\left(\begin{array}{c}p\\ 1-8p\end{array}\right){\mathrm{ms}}^{-1}$, where $p$ and $q$ are constants.

The particle's displacement after 9 seconds is $\left(\begin{array}{c}-22.5\\ -81\end{array}\right)\mathrm{m}$.

Show that

$\left(\begin{array}{c}-22.5\\ -81\end{array}\right)=4.5\left(\begin{array}{c}p+2q-1\\ 1-8p+q\end{array}\right)$

Hence find the values of $p$ and $q$.

A train leaves station $O$ from rest with constant acceleration $(0.3\mathbf{i}+0.7\mathbf{j})\mathrm{m}{\mathrm{s}}^{-2}.$

After 80 seconds it passes through, but does not stop at, station $A$.

Find the displacement of the train from station $O$ when it passes through station $A$.

Find the velocity of the train as it passes through station $A$.

When the train passes through station $A$ its acceleration changes to $(0.5\mathbf{i}+0.3\mathbf{j})\mathrm{m}{\mathrm{s}}^{-2}.$

After another 180 seconds the train passes through station $B$.

Find the distance the train travels from station $A$ from station $B$.

A ball is thrown from the top of a tall building with an initial velocity of  $\left(\begin{array}{c}0.8\\ 0.2\end{array}\right)\mathrm{m}{\mathrm{s}}^{-1}$.

Find the velocity of the ball 5 seconds after it is thrown.

Hence, find the speed of the ball 5 seconds after it is thrown.

A particle experiences a constant acceleration of $\left(3p\mathbf{i}+2\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-2}$ where $p$ is a constant.

In 7 seconds the particle travels $(-91\mathbf{i}+7\mathbf{j})\mathrm{m}$.

Given that the particle’s velocity after the 7 seconds is $\left(4p\mathbf{i}+8\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-1}$ find the value of the constant $p$.

Hence find the exact magnitude of the acceleration during this motion.

A horse runs across a large area of land. It starts to gallop with constant acceleration $(0.6\mathbf{i}+0.4\mathbf{j}){\mathrm{ms}}^{-2}$.

After 12 seconds of galloping the horse has velocity $(8\mathbf{i}+10\mathbf{j}){\mathrm{ms}}^{-1}$.

Find the displacement of the horse at the end of the 12 second period.

Find the change in speed of the horse between the start and end of its 12 second gallop.

Two stones are slid across a large icy pond.  Both are released from rest at the origin.

The first stone experiences a constant acceleration of $\left(3\mathbf{i}-\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-2}$.  

The second stone experiences a constant acceleration of $\left(-2\mathbf{i}+\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-2}$. 

Find the distance between the two stones after 8 seconds.

A ball is thrown from the top of a tall building with a velocity of $\left(\begin{array}{c}3.5\\ 6\end{array}\right)\mathrm{m}{\mathrm{s}}^{-1}$.

Find the speed of the ball 3 seconds after it is thrown.

The ball first strikes the ground after 6 seconds.

Find the displacement of the ball relative to its starting point when it first strikes the ground.

A particle moves with a constant acceleration of $\left(\begin{array}{c}2q+1\\ q-1\end{array}\right)\mathrm{m}{\mathrm{s}}^{-2}$.

The particle has initial velocity $\left(\begin{array}{c}5.2\\ 1-2p\end{array}\right)\mathrm{m}{\mathrm{s}}^{-1}$and 6 seconds later has the particle has velocity $\left(\begin{array}{c}27p+4\\ 5.2\end{array}\right)\mathrm{m}{\mathrm{s}}^{-1}$.

Given that $p$ and $q$ are constants, find the values of $p$ and $q$.

Starting from rest, a toy boat sails across a pond such that for the first $10\mathrm{seconds}$ of its motion it has constant acceleration $\left(0.1\mathbf{i}+0.3\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-2}$.

It then sails with a constant velocity until it reaches the other side of the pond, 6 seconds later.

Find the distance between the toy boat’s starting position and its position once it has reached the other side of the pond.

Give your answer to three significant figures.

Briefly explain why your answer to part (a) is not necessarily the length nor width of the pond.

A particle travels $(16\mathbf{i}+96\mathbf{j})\mathrm{m}$ in 8 seconds with a constant acceleration of $(a\mathbf{i}-4\mathbf{j}){\mathrm{ms}}^{-2}$.

Given that the particle’s velocity after the 8 seconds is $(14\mathbf{i}+b\mathbf{j}){\mathrm{ms}}^{-1}$ find the values of the constants $a$ and $b$.

Find the velocity of the particle at the start of these 8 seconds.

Two stones slide across a large icy pond.

The first stone is released from rest at the origin with constant acceleration $(2\mathbf{i}+3\mathbf{j})\mathrm{m}{\mathrm{s}}^{-2}.$

The second stone is released from rest from a displacement of $\left(50\mathbf{i}-100\mathbf{j}\right)\mathrm{m}$ relative to the origin, with constant acceleration  $(\mathbf{i}+5\mathbf{j})\mathrm{m}{\mathrm{s}}^{-2}$.

Find the distance between the two stones after 5 seconds.

Show that the two stones collide after 10 seconds.

A particle travelling with constant acceleration takes 6 minutes to travel  $\left(\begin{array}{c}5.94\\ 13.86\end{array}\right)\mathrm{km}$.

The particle’s velocity at the start of the 6 minute period is one tenth of its velocity at the end of the 6 minute period.

Find the acceleration of the particle.

A football is kicked from the top of a hill and its motion is modelled as moving in a 2D vertical plane under the force of gravity only.

The ball is kicked such that its initial velocity is $(15\mathbf{i}+24\mathbf{j})\mathrm{m}{\mathrm{s}}^{-1}$.

Given that the top of the hill is 6 m vertically above ground level, find the time it takes the football to first hit the ground.

Find the horizontal distance covered by the football until it first hits the ground.

Find the speed with which the football first hits the ground.

A train leaves station $O$ from rest with constant acceleration $(0.5\mathbf{i}+0.2\mathbf{j})\mathrm{m}{\mathrm{s}}^{-2}$.

60 seconds later it passes through, but does not stop at, station $A$.

At station $A$, the train's acceleration changes to $(0.8\mathbf{i}+0.1\mathbf{j})\mathrm{m}{\mathrm{s}}^{-2}$.

90 seconds after passing through station $A$, the train passes through station $B$.

Find the total distance the train travels on its journey from station $O$, through $A$, to station $B$.

Find the average acceleration of the train between station O and station B.

It takes three minutes for a particle to travel $\left(\begin{array}{c}1.62\\ 2.16\end{array}\right)\mathrm{km}$ with constant acceleration.

The particle’s velocity at the start of the three minutes is half of its velocity at the end.

Find the exact magnitude of the acceleration in metres per second squared.

A small ball is kicked from the top of a hill above horizontal ground. Its motion modelled as moving in a vertical 2D plane under gravity.

Its initial velocity is $(12\mathbf{i}+7\mathbf{j})\mathrm{m}{\mathrm{s}}^{-1}$ and it first hits the ground with velocity $(12\mathbf{i}-20\mathbf{j})\mathrm{m}{\mathrm{s}}^{-1}$.

Find the distance between the point from which the ball was kicked and the point at which it first hits the ground.

Starting from rest, a toy boat sails across a pond such that for the first 15 seconds of its motion it has constant acceleration $(0.12\mathbf{i}+0.05\mathbf{j})\mathrm{m}{\mathrm{s}}^{-2}$.

It then decelerates uniformly until it comes to rest 8 seconds later on the other side of the pond.

Find the distance between the toy boat’s initial and final positions.

A small ball is thrown from the top of a tall building with velocity  $\left(\begin{array}{c}4\\ 8.5\end{array}\right)\mathrm{m}{\mathrm{s}}^{-1}$.

Find the speed of the ball 2 seconds after it is thrown.

The ball first strikes the ground after 5 seconds.

Find the distance between the point where the ball first strikes the ground and the ball's starting point at the top of the building.

A particle travels $(464\mathbf{i}-272\mathbf{j})\mathrm{m}$ in 16 seconds with a constant acceleration given by

$\left(-\frac{p}{q}\mathbf{i}+\frac{2p}{q}\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-2}$

where $p$ and $q$ are positive non-zero constants.

Given that the particle’s velocity after the 16 seconds is

$\left(\right(2q+3)\mathbf{i}+(p+5\left)\mathbf{j}\right)\mathrm{m}{\mathrm{s}}^{-1}$ 

find the values of $p$ and $q$.

[In this question, $\mathbf{i}$ and $\mathbf{j}$ are horizontal unit vectors and position vectors are given relative to a fixed origin $O$]

Two stones slide across a large icy pond.

At time $t=0$ seconds, the first stone is located at the point with position vector $\left(2\mathbf{i}-8\mathbf{j}\right)\mathrm{m}$ with an initial velocity $(5.4\mathbf{i}+7.2\mathbf{j})\mathrm{m}{\mathrm{s}}^{-1}$. It moves with constant acceleration $(0.4\mathbf{i}+0.6\mathbf{j})\mathrm{m}{\mathrm{s}}^{-2}.$

At time $t=0$ seconds, the second stone is located at the point with position vector $(50\mathbf{i}+40\mathbf{j})\mathrm{m}$ with initial velocity $(-4.6\mathbf{i}-2.8\mathbf{j})\mathrm{m}{\mathrm{s}}^{-1}$. It moves with constant acceleration $(1.4\mathbf{i}+1.6\mathbf{j})\mathrm{m}{\mathrm{s}}^{-2}$.

Determine the distance from the origin of the stones when they first collide.

A particle passes a fixed point $O$ at time $t=0$ seconds. $T$ seconds later the particle has velocity $\left(\begin{array}{c}2T+4\\ 1-T\end{array}\right)\mathrm{m}{\mathrm{s}}^{-1}$.

The acceleration of the particle throughout this motion is $\left(\begin{array}{c}T-3\\ T-9\end{array}\right)\mathrm{m}{\mathrm{s}}^{-2}$.

The displacement of the particle relative to $O$, $T$ seconds after it passes $O$ is $\left(\begin{array}{c}4T\\ T\end{array}\right)\mathrm{m}$. 

Show that the initial velocity of the particle is $\left(\begin{array}{c}-10\\ 8\end{array}\right)\mathrm{m}{\mathrm{s}}^{-1}$.

A train leaves station $O$ from rest with constant acceleration $(0.4\mathbf{i}+0.1\mathbf{j})\mathrm{m}{\mathrm{s}}^{-2}$.

2.5 minutes later it passes, but does not stop at, station $A$. At this point its acceleration changes to $(0.2\mathbf{i}+0.3\mathbf{j})\mathrm{m}{\mathrm{s}}^{-2}$.

5 minutes after passing through $A$ the train passes through station $B$.

Find the average velocity of the train between station $O$ and station $B$.