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

The velocity, $v\mathrm{m}{\mathrm{s}}^{-1}$, of a particle moving in a straight line at time $t$ seconds is given by $v=4t-t^2$ for $0\le t\le 5$.

Verify that the particle is instantaneously at rest when $t=0\mathrm{and}t=4.$ 

Sketch a velocity-time graph for the motion of the particle during the interval $0\le t\le 5$. Label the axes intercepts, any maximums or minimums, and the final velocity at $t=5$.

Show that:

(i) the particle travels a distance of  $\frac{32}{3}$m between $t=0\mathrm{and}t=4$.

(ii) the particle travels a distance of   $\frac{7}{3}$m between $t=4\mathrm{and}t=5$.

Find the total distance travelled by the particle between $t=0$ and $t=5$.

Explain why the distance between the position of the particle at $t=0$ and the position of the particle at $t=5$ is  $\frac{25}{3}$m.

A particle $P$ moves along a straight line.

At time $t$ seconds, the velocity $v$ ms⁻¹ of $P$ is modelled as

$v=10t-t^2-kt\ge 0$

where $k$ is a constant.

Find the acceleration of $P$ at time $t$ seconds.

The particle $P$ is instantaneously at rest when $t=6$.

Find the other value of $t$ when $P$ is instantaneously at rest.

Find the total distance travelled by $P$ in the interval $0\le t\le 6$

A car is travelling along a straight horizontal motorway and passes a junction at time $t=0$ seconds.

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

$s=18t^2-t^3$

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

(ii) Find the time, other than at $t=0$, that the model shows the car passing the same junction.

(i) Find an expression for the velocity, $v{\mathrm{ms}}^{-1},$ of the car at time t seconds.

(ii) Find the time, other than at $t=0$, that the model shows the car is instantaneously at rest.

A particle moving along a straight line has velocity $v\mathrm{m}{\mathrm{s}}^{-1}$, at time $t$ seconds, and its motion is described the equation

$v=t^2-4t+4$

(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

$s=0.4t^2+3.5t$

where $s$ m is their displacement from the starting point at time $t$ 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

$v=\frac{1}{10}t\left(36-t\right)t\ge 0$

where $v\mathrm{m}{\mathrm{s}}^{-1}$ 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 $t=0$ seconds.

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

 $a=40+6t-t^2$                        $t\ge 0$

Find the acceleration of the rocket after 9 seconds.

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

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

In a cheese-rolling competition, a cylindrical block of cheese starts from rest and then rolled down a hill. Its acceleration, $a{\mathrm{ms}}^{-2}$ , is modelled by the equation

 $a=1+0.1t$                     $0\le t\le 20$

where $t$ 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.

The velocity, $v$ ms⁻¹ , of a particle moving in a straight line at time $t$ seconds is given by $v=12t-2t^2$  for $0\le t\le 10$.

Sketch a velocity-time graph for the motion of the particle during the interval $0\le t\le 10$. Label the axes intercepts and any maximum or minimum points.

Show that the total distance travelled by the particle is $\frac{632}{3}\mathrm{m}$.

You must show your full working when evaluating the integrals.

A particle moving along a straight line has velocity, $v\mathrm{m}{\mathrm{s}}^{-1}$, at time $t$ seconds according to the equation 

$v=t^2-6t+8$

Find the times at which the particle is instantaneously stationary.

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

A fixed point $O$ lies on a straight line.

A particle $P$ moves along the straight line.

At time t seconds, $t\ge 0$, the distance, $s$ metres, of $P$ from $O$ is given by

$s=\frac{1}{3}{t}^3-\frac{5}{2}{t}^2+6t$

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

Find the total distance travelled by $P$ in the interval $0\le t\le 4$.

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

Solutions relying entirely on calculator technology are not acceptable.

A fixed point $O$ lies on a straight line.

A particle $P$ moves along the straight line such that at time $t$ seconds, $t\ge 0$, after passing through $O$, the velocity of $P$, $v$ ms^–1, is modelled as

$v=15–t^2–2t$

Verify that $P$ comes to instantaneous rest when $t=3$.

Find the magnitude of the acceleration of $P$ when $t=3$.

Find the total distance travelled by $P$ in the interval $0\le t\le 4$.

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

$s=1.8t+0.75t^2-0.02t^3$

where $s$ m is the displacement from the starting point at time $t$ 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 

$v=\frac{1}{w}{t}^2\left(60-t\right)$ 

where $vm{s}^{-1}$ is the velocity of the go kart at time $t$ seconds and $w$ is a constant.

Given the maximum speed of the go kart is $32{\mathrm{ms}}^{-1}$, find the value of $w$ 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 $t=0$ seconds, from ground level with an acceleration of $56\mathrm{m}{\mathrm{s}}^{-2}$.

The rocket’s acceleration is then modelled by the equation                          

$a=56+t-t^2$                              $t\ge 0$ 

(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 $0\mathrm{m}{\mathrm{s}}^{-1}$.

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 $A$ and point $B$.

The velocity-time graph below shows the motion of a child on the zip-wire as they move from $A$at $t=0$ and reach point $B$ at $t=16$.

For $0\le t\le 4$, the graph has the equation $v=5\sqrt{t}$ where $v\mathrm{m}{\mathrm{s}}^{-1}$ is the velocity at time $t$ seconds. 

(i) Find the distance between point $A$and point $B$.

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

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

A bullet train has a maximum acceleration of $0.72\mathrm{m}{\mathrm{s}}^{-2}$.

A particular bullet train leaves a station at time $t=0$ seconds and its displacement, $s\mathrm{m}$, from the station is modelled using the equation 

$s=\frac{3}{200}{t}^{3}0\le t\le 8$

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, $a\mathrm{m}{\mathrm{s}}^{-2}$, of a particle moving in a straight line at time $t$ seconds is given by $a=4t-7$ for  $0\le t\le 6$.

Initially the velocity of the particle is $3\mathrm{m}{\mathrm{s}}^{-1}$.

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 $400\mathrm{m}$ sprint is aiming to run according to the model

$s=0.002\left(4000t+50t^2-t^3\right)$

where $s$  is the displacement from the starting point at time $t$ 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 $t=0$ seconds.

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

$s=0.4t(2t^2-4t+3)$

Show that the model indicates that the car never returns to the service station it passes at $t=0$ seconds. Show your reasoning carefully.

Show that the car's speed is decreasing for the first $\frac{2}{3}$ 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

$v=\{\begin{array}{l}0.25(t^3-20t^2+100t)0\le t\le p\\ 12t\ge p\end{array}$

where $v\mathrm{m}{\mathrm{s}}^{-1}$ is the velocity of the go kart at time $t$ seconds.

Given that $p$ is an integer, find the value of $p$.

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

A home-made rocket is launched from rest at ground level at time $t=0$ seconds. The rocket travels only vertically. Its acceleration is initially $64\mathrm{m}{\mathrm{s}}^{-2}$ and is modelled by the equation

$a=64+12t-t^{2}t\ge 0$

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, $a\mathrm{m}{\mathrm{s}}^{-2}$ , is modelled by the functions

 $a\left(t\right)=\{\begin{array}{l}0.2t0\le t\le 15\\ 9-t15<t\le A\end{array}$ 

where $t$ is the time in seconds and $A$ is a constant.

The block of cheese comes to rest when its acceleration is $-9\mathrm{m}{\mathrm{s}}^{-2}.$

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

A high-speed train leaves a station at time $t=0$ seconds and its displacement, $s$metres, from the station is modelled using the equation

 $s=\frac{1}{p}{t}^{q}0\le t\le 12$

where $p$ and $q$ are constants.

In the first 10 seconds after the train leaves the station, the average velocity is  $\frac{5}{12}\mathrm{m}{\mathrm{s}}^{-1}$ and the average acceleration is  $\frac{1}{6}\mathrm{m}{\mathrm{s}}^{-2}$.

By first finding the values of $p$ and $q$, find an expression for the acceleration of the high-speed train for $0\le t\le 12.$

The acceleration, $a\mathrm{km}{\mathrm{h}}^{-2}$, of a particle moving in a straight line at time $t$ hours is given by $a=\frac{1}{5}(t-11)$ for  $0\le t\le 24.$

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

Show that the velocity, $v\mathrm{km}{\mathrm{h}}^{-1}$, of the particle at time $t$ hours can be written as

 $v=\frac{1}{10}(t^2-22t+k)$

where $k$ 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.