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

A particle moves along a straight line.

At time seconds, the velocity ms⁻¹ of is modelled as

where is a constant.

Find the acceleration of at time seconds.

The particle is instantaneously at rest when .

Find the other value of when is instantaneously at rest.

Find the total distance travelled by in the interval

A car is travelling along a straight horizontal motorway and passes a junction at time seconds.

The car’s displacement, metres, from the junction is then modelled by the equation

(i) Find the displacement of the car from the junction after 3 seconds.

(ii) Find the time, other than at , that the model shows the car passing the same junction.

(i) Find an expression for the velocity, of the car at time t seconds.

(ii) Find the time, other than at , that the model shows the car is instantaneously at rest.

A particle moving along a straight line has velocity , at time seconds, and its motion is described the equation

(i) Write down the initial velocity of the particle.

(ii) Find the time at which the particle is instantaneously stationary.

Show that the acceleration of the particle is negative for the first 2 seconds of its motion.

An athlete training for the 100 m sprint is aiming to run according to the model

where m is their displacement from the starting point at time seconds.

Find, according to the model, the time it should take the athlete to complete the 100 m sprint, giving your answer to one decimal place.

Assuming the athlete's motion follows the model, show that their acceleration is constant.

A go-kart manufacturer is testing out a new go-kart model on a straight horizontal road.

Starting from rest, the velocity of the go-kart is modelled by the equation

where is the velocity at time t seconds.

Find the maximum velocity of the go kart and the time at which this occurs.
Justify that this is a maximum.

The go kart does not move backwards at any point during the test.
Find the time it takes to complete the test.

A home-made rocket is launched from rest at ground level at time  seconds.

The acceleration of the rocket, measured in metres per square second, is modelled by the equation

Find the acceleration of the rocket after 9 seconds.

Find an expression for the velocity of the rocket at time .

Find an expression for the displacement of the rocket at time .

A particle moving along a horizontal path has acceleration  at time  seconds modelled by the equation

The particle has a velocity of at time .

Find an expression for the velocity of the particle at time  seconds.

(i) Find the time at which the velocity of the particle is zero.

(ii) Hence write down the times between which the particle has a positive velocity.

In a cheese-rolling competition, a cylindrical block of cheese starts from rest and then rolled down a hill. Its acceleration, , is modelled by the equation

where is the time in seconds.

The block of cheese reaches the bottom of the hill after 20 seconds.

Find the velocity of the block of cheese when it reaches the bottom of the hill.

Show that the distance down the hill, as travelled by the block of cheese, is 330 m to two significant figures.

A high-speed train has a maximum acceleration of  which, from rest, takes 20 seconds to reach.

The train leaves a station at  seconds and its displacement, , from the station is modelled using the equation

 where is a constant.

Find an expression for the velocity of the high-speed train for .

(i) Find an expression for the acceleration of the high-speed train for .

(ii) Hence find the value of the constant , assuming that the train reaches its maximum acceleration in the quickest time possible.

Find the minimum distance of track needed in order for the high-speed train to reach its maximum acceleration.

The velocity, ms⁻¹ , of a particle moving in a straight line at time seconds is given by   for .

Sketch a velocity-time graph for the motion of the particle during the interval . Label the axes intercepts and any maximum or minimum points.

Show that the total distance travelled by the particle is .

You must show your full working when evaluating the integrals.

A particle moving along a straight line has velocity, , at time  seconds according to the equation 

Find the times at which the particle is instantaneously stationary.

Determine how far the particle moves while its velocity is negative.

A particle moving along a straight line has velocity,, at time seconds according to the equation

Find the times at which the particle is instantaneously at rest.

Find the times between which the acceleration of the particle is negative.

A fixed point lies on a straight line.

A particle moves along the straight line.

At time t seconds, , the distance, metres, of from is given by

Find the acceleration of at each of the times when is at instantaneous rest.

Find the total distance travelled by in the interval .

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A fixed point lies on a straight line.

A particle moves along the straight line such that at time seconds, , after passing through , the velocity of , ms^–1, is modelled as

Verify that comes to instantaneous rest when .

Find the magnitude of the acceleration of when .

Find the total distance travelled by in the interval .

An athlete training for the 200 m sprint is aiming to run according to the model 

where m is the displacement from the starting point at time seconds.

Find the time the athlete should be expected to finish the 200 m sprint in.

Find the average acceleration that the athlete would achieve when sprinting the 200 m, according to this model.

A go kart manufacturer is testing out a new model on a straight horizontal road.

Starting from rest, the velocity of the go kart is modelled by the equation 

where is the velocity of the go kart at time seconds and is a constant.

Given the maximum speed of the go kart is , find the value of and the time at which the go kart reaches its maximum speed.

(i) Find the maximum acceleration of the go kart according to the model.

(ii) Justify that your answer to part (i) is a maximum.

A home-made rocket is launched from rest, at time  seconds, from ground level with an acceleration of .

The rocket’s acceleration is then modelled by the equation                          

(i) Find an expression for the velocity of the home-made rocket.

(ii) Other than at launch, find the time when the velocity of the rocket is .

Find the greatest height the rocket reaches, giving your answer in kilometres to three significant figures.

A zip-wire in a children's park runs between point and point .

The velocity-time graph below shows the motion of a child on the zip-wire as they move from at and reach point at .

For , the graph has the equation  where is the velocity at time seconds. 

(i) Find the distance between point and point .

(ii) Find the distance between the child and point when the child comes to rest.

Find the acceleration of the zip-wire after 1 second.

A bullet train has a maximum acceleration of .

A particular bullet train leaves a station at time seconds and its displacement, , from the station is modelled using the equation 

Show that it takes 8 seconds for the bullet train to reach its maximum acceleration.

After reaching its maximum acceleration the bullet train continues to accelerate at 0.72 ms^-2 until its velocity reaches its maximum of 75 ms^-1.

Find the length of time between the train reaching its maximum acceleration and when it reaches its maximum velocity of 75 ms^-1.

Once reaching its maximum velocity, the bullet train continues at this velocity for 10 minutes.

Find the displacement of the train from the station after 10 minutes, giving your answer in kilometres to 3 significant figures.

The acceleration, , of a particle moving in a straight line at time  seconds is given by  for  .

Initially the velocity of the particle is .

Find the time(s) when the particle is instantaneously at rest.

In this question part you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Find the exact total distance travelled by the particle in the first 6 seconds of motion.

Show your full method clearly.

An athlete training for the  sprint is aiming to run according to the model

where  is the displacement from the starting point at time seconds.

To help the athlete keep pace, markers are put every 100 m along the track, as well as at the finish line.

Considering the model and the scenario, find the times that the athlete should pass the 100 m, 200 m and 300 m markers and the finish line marker.

Find the average acceleration, according to the model, for the last 100 m of the sprint.

Interpret this result in the context of the model.

A car is travelling along a straight horizontal motorway and passes a service station at time seconds.

The car’s displacement, metres, from the service station is then modelled by the equation

Show that the model indicates that the car never returns to the service station it passes at seconds. Show your reasoning carefully.

Show that the car's speed is decreasing for the first  seconds after passing the service station.

Show your full reasoning.

A particle moves in a straight horizontal line.

Starting from rest, the velocity of the particle is modelled by the equation

where is the velocity of the go kart at time seconds.

Given that is an integer, find the value of .

Find the maximum and minimum velocities of the particle in the first seconds of its motion.

A home-made rocket is launched from rest at ground level at time seconds. The rocket travels only vertically. Its acceleration is initially and is modelled by the equation

Assuming the rocket lands back at ground level, find the total distance travelled by the rocket and the total time it spends in the air.

In a cheese-rolling competition, a cylindrical block of cheese is rolled down a hill, and then continues to roll along a horizontal surface.

Its acceleration, , is modelled by the functions

where is the time in seconds and is a constant.

The block of cheese comes to rest when its acceleration is

Find the distance the block of cheese rolls before it comes to rest.

A high-speed train leaves a station at time seconds and its displacement, metres, from the station is modelled using the equation

where and are constants.

In the first 10 seconds after the train leaves the station, the average velocity is   and the average acceleration is  .

By first finding the values of and , find an expression for the acceleration of the high-speed train for

The acceleration, , of a particle moving in a straight line at time hours is given by for 

After 24 hours the particle has returned to where it started.

Show that the velocity, , of the particle at time hours can be written as

where is a constant to be found.

For this part you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Find the exact total distance travelled by the particle in the first 24 hours of motion.