Constant Acceleration in 1D (Edexcel A Level Maths: Mechanics): Exam Questions

The motion of a particle is described by the velocity-time graph in Figure 1.

Work out the acceleration for the first 6 seconds of the particle’s motion.

Work out the distance covered by the particle in the last 10 seconds of its motion.

The particle travels a distance of 280 m whilst it has zero acceleration.

Find the value of $T$.

The point $A$ is 1.8m vertically above horizontal ground.

At time $t=0$, a small stone is projected vertically upwards with speed $U$ ms⁻¹ from the point $A$.

At time $t=T$ seconds, the stone hits the ground.

The speed of the stone as it hits the ground is 10 ms⁻¹.

In an initial model of the motion of the stone as it moves from $A$ to where it hits the ground

• the stone is modelled as a particle moving freely under gravity

• the acceleration due to gravity is modelled as having magnitude 10ms⁻²

Using the model, find the value of $U$.

Using the model, find the value of $T$.

Suggest one refinement, apart from including air resistance, that would make the model more realistic.

In reality the stone will not move freely under gravity and will be subject to air resistance.

Explain how this would affect your answer to part (a).

A car is being tested for safety by using a computer‑controlled car that is accelerated along a horizontal track and crashed into a wall.

The horizontal track is 750 m long, and the wall is at the end of the track.

During a particular crash test, the car starts from rest and has a constant acceleration of $1.5{\mathrm{ms}}^{-2}$.

Find the maximum speed, in metres per second, that the car can reach along the track.

(i) Calculate the distance from the wall a car should be placed such that when starting from rest, it will crash into the wall with a speed of $27\mathrm{m}{\mathrm{s}}^{-1}$?

(ii) In this case, calculate the length of time it will take for the car to reach the wall.

A ball is projected vertically upwards from the ground with a velocity of $5.8{\mathrm{ms}}^{-1}$.

Find

(i) the maximum height the ball reaches above the ground,

(ii) and the time taken to reach the maximum height.

Train $A$ leaves a station, starting from rest, with constant acceleration. After 85 seconds it is passed by train $B$.

Train $B$ left the same station, also starting from rest, 35 seconds after train $A$.

Train $B$ moves with constant acceleration $1.4{\mathrm{ms}}^{-2}$ throughout its motion.

Train $B$ passes train $A$ at point $X$.

Calculate the distance between the station and point $X$.

Calculate the acceleration of train $A$, giving your answer to three significant figures.

A particle is projected vertically upwards from ground level with a velocity of $35.6{\mathrm{ms}}^{-1}$.

Find the length of time for which the particle is at least 15 m above the ground.

A train leaves station $O$ from rest with constant acceleration $0.12{\mathrm{ms}}^{-2}$.

190 seconds later it passes a signal at which point the train decelerates uniformly at $0.18{\mathrm{ms}}^{-2}$ until coming to rest at station $X$ .

Find the distance between station $O$ and station $X$.

To crash test cars a computer‑controlled car is accelerated along a horizontal track and crashed into a wall. The maximum length of track available is $0.8\mathrm{km}$.

During a crash test, a car starts from rest and has constant acceleration.

In one test a car is driven at the wall with constant acceleration $1.6{\mathrm{ms}}^{-2}$.

Find the maximum speed, in kilometres per hour, with which it could hit the wall.

The standard setting for testing is a constant acceleration of $1.2{\mathrm{ms}}^{-2}$, but this can be varied up or down by 40% prior to a test being carried out.

Determine if it is possible to crash test a car at a speed of $200{\mathrm{kmh}}^{-1}$.

Use the constant acceleration equations 

$v=u+at$    and    $s=ut+\frac{1}{2}a{t}^2$

to show that

 $v^2=u^2+2as$

A stone is projected vertically downwards from the top of a cliff with initial speed $0.3\mathrm{m}{\mathrm{s}}^{-1}.$ The stone hits the sea below after 3.2 seconds.

Calculate the height of the cliff.

At time $t=0$, a small stone is thrown vertically upwards with speed 14.7 ms⁻¹ from a point $A$.

At time $t=T$ seconds, the stone passes through $A$, moving downwards.

The stone is modelled as a particle moving freely under gravity throughout its motion.

Using the model, find the value of $T$.

Using the model, find the total distance travelled by the stone in the first 4 seconds of its motion.

State one refinement that could be made to the model, apart from air resistance, that would make the model more realistic.

A small stone is projected vertically upwards with speed 39.2 ms^–1 from a point $O$.

The stone is modelled as a particle moving freely under gravity from when it is projected until it hits the ground 10 s later.

Using the model, find the height of $O$ above the ground.

Using the model, find the total length of time for which the speed of the stone is less than or equal to 24.5 ms^-1.

State one refinement that could be made to the model that would make your answer to part (a) more accurate.

A ball is projected vertically upwards from ground level.

1.6 seconds after reaching its maximum height, the ball hits the ground.

Find

(i) the maximum height the ball reached,

(ii) the velocity with which it was projected.

A car travelling along a horizontal road passes a point $A$ with velocity $21{\mathrm{ms}}^{-1}$ and constant acceleration $0.2{\mathrm{ms}}^{-2}$.

Point $B$ is $1.5\mathrm{km}$ from point $A$. When the car reaches point $B$ it decelerates uniformly at $3.1{\mathrm{ms}}^{-2}$ until it comes to rest.

Find the distance the car travels from the moment it starts to decelerate until it comes to rest.

Train $A$ leaves a station from rest with constant acceleration $0.12{\mathrm{ms}}^{-2}$.

After 40 seconds have elapsed, train $B$ leaves the station from rest with constant acceleration $0.2{\mathrm{ms}}^{-2}$, travelling along the same track as train $A$.

Find the length of time from when train $B$ leaves the station to when it catches up with train $A$.

Give your answer to three significant figures.

Find the distance, in kilometres, that both trains have travelled when train $A$ catches up with train $B$. Give your answer to three significant figures.

A train leaves station $O$ from rest with a constant acceleration of $0.2{\mathrm{ms}}^{-2}.$

After $125\mathrm{seconds}$ it passes a signal at which point the train decelerates uniformly until coming to rest at station $X$, 75 seconds later.

Find the distance between station $O$ and station $X$.

The train then leaves station $X$  but travels in the opposite direction with constant acceleration $0.1{\mathrm{ms}}^{-2}$.

The train does not stop at station $O$ but $300\mathrm{seconds}$ after leaving station $X$  it passes a sign indicating that station $Y$  is 850 m away. At this point the train starts to decelerate uniformly so that it comes to rest at station $Y$.

Find the distance between station $O$ and station $Y$.

A particle moves with constant acceleration, $a\mathrm{m}{\mathrm{s}}^{-2}$, such that its initial velocity is $u\mathrm{m}{\mathrm{s}}^{-1}$ and $t$ seconds later its velocity is $v\mathrm{m}{\mathrm{s}}^{-1}$.

Use calculus to show that the displacement of the particle, $s$ m, from its initial position is given by

$s=vt-\frac{1}{2}a{t}^2$

Show clear working for each stage of your solution.

A car travels along a horizontal road and passes a point $A$ with velocity $17.2{\mathrm{ms}}^{-1}$ and constant acceleration $0.4{\mathrm{ms}}^{-2}$.

Point $B$ is $0.8\mathrm{km}$ from point $A$. When the car reaches point $B$ it starts decelerating at a constant rate of $2.75\mathrm{m}{\mathrm{s}}^{-2}$ until it comes to rest at point $C$.

Find the time taken by the car to travel from point $A$ to point $C$.

Give your answer to one decimal place.

A train leaves station $A$ from rest, heading in the direction of station $B$ with a constant acceleration of magnitude $0.2{\mathrm{ms}}^{-2}$ .

At the same time, another train leaves station $B$ from rest, heading in the direction of station $A$ with a constant acceleration of magnitude $0.16{\mathrm{ms}}^{-2}.$

The distance between station $A$and station $B$ is $8.2\mathrm{km}$.

Determine the distance from station $A$ to the point at which the two trains meet.

Give your answer in kilometres to 2 decimal places.

Figure 1 shows a velocity time graph for the motion of two particles.

The motion of particle $A$ is shown by the solid line, and the motion of particle $B$ is shown by the dotted line.

Find the value of $T_1$, giving your answer to three significant figures.

Find the value of $T_2$, giving your answer to three significant figures.

Two trains leave station $O$, both from rest at the same time, in opposite directions.

The first train travels with constant acceleration of magnitude $0.15\mathrm{m}{\mathrm{s}}^{-2}$.

The second train travels with constant acceleration of magnitude $0.24\mathrm{m}{\mathrm{s}}^{-2}$ until it reaches a signal 210 seconds later. At this point it starts to decelerate uniformly until coming to rest at station $X$, 60 seconds later.

After a 2-minute wait at station $X$, the second train leaves in the opposite direction (back towards $O$) with constant acceleration of magnitude $0.8\mathrm{m}{\mathrm{s}}^{-2}.$

Find the distance between the two trains 10 minutes after they both left station $O$.

A ball is projected vertically upwards from ground level with speed $29.3\mathrm{m}{\mathrm{s}}^{-1}$.

At the same time, a second ball is projected vertically downwards from a height of 150 m above ground level, directly above the first ball, with speed $8.2\mathrm{m}{\mathrm{s}}^{-1}$.

Find

(i) the time it takes the two balls to collide,

(ii) the height above the ground at which this collision occurs.

Find the speed of each ball at the point when they collide, and state their direction of motion when the collision occurs.