Variable Acceleration - 1D (AQA A Level Maths: Mechanics)

Exam Questions

4 hours32 questions
1a
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1 mark

A particle moving in a straight line has displacement, s m, from its initial position at time, t seconds, given by the equation                        

               s equals 3 t squared plus 4 t

Find the displacement of the particle after 12 seconds.

1b
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2 marks
(i)
Find, by differentiating, an expression for the velocity after t seconds.

(ii)
Find the velocity of the particle after 8 seconds.

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2a
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2 marks

A particle moving in a straight line has velocity, v space straight m space straight s to the power of negative 1 end exponent, at time, t seconds, given by the equation

               v equals 0.2 t squared minus 0.1 t

Find the time at which the velocity of the particle reaches 1 space straight m space straight s to the power of negative 1 end exponent .

2b
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2 marks
(i)
Find, by differentiating, an expression for the acceleration after t space seconds.

(ii)
Find the acceleration of the particle after 6 seconds.

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3a
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1 mark

A particle moving in a straight line has acceleration, a space straight m space straight s to the power of negative 2 end exponent, at time, t seconds, given by the equation

                        a equals 6 t minus 2

Find the time at which the particle is accelerating at 10 space straight m space straight s to the power of negative 2 end exponent.

3b
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3 marks

After 5 seconds the velocity of the particle is 68 m s−1.

(i)
Use integration to find an expression for the velocity after t seconds.

(ii)
Find the velocity after 8 seconds.

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4a
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2 marks

A particle moving in a straight line has velocity, v space straight m space straight s to the power of negative 1 end exponent, at time, t seconds, given by the equation

               v equals 8 t cubed minus 6 t squared

Other than at t equals 0, find the time when the particle is stationary.

4b
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3 marks
(i)
Find an expression, by integrating v with respect to t, for the displacement of the particle from its initial position, after t seconds.

(ii)
Find the times at which the particle is at its initial position.

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5a
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2 marks

The velocity, v space straight m space straight s to the power of negative 1 end exponent, of a particle moving in a straight line at time t seconds can be found using the following

v equals open curly brackets table row cell open parentheses t minus 4 close parentheses open parentheses t plus 1 close parentheses space space space space 0 less or equal than t less or equal than 6 end cell row cell 14 space space space space space space space space space space space space space space space space space space space space space space space space space space space t greater or equal than 6 end cell end table close curly brackets

(i)
Find the initial speed of the particle.

(ii)
Write down the acceleration for t space greater or equal than space 6.
5b
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2 marks

Find, by differentiation, an expression for the acceleration for 0 space less or equal than space space t space space less or equal than space 6.

5c
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3 marks

Use integration to show that the displacement of the particle from its initial position for 0 less or equal than t less or equal than 6 space is given by

s equals space 1 third t cubed minus 3 over 2 t squared minus 4 t

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6a
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3 marks

A particle is moving in a straight line and at time t seconds has acceleration, a space straight m space straight s to the power of negative 2 end exponent, where a space equals space 12 t space minus space 12 t squared space plus 10.

Show by integrating twice that the displacement, s m, of the particle from a fixed point O , is given by

s = 2t3t4 + 5t2 + ct + d

where c and d are constants.

6b
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2 marks

Given that the particle started from rest at the point O, write down the values of c and d, and find the displacement of the particle after 5 seconds.

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7a
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4 marks

The velocity, v space straight m space straight s to the power of negative 1 end exponent, of a particle moving in a straight line at time t seconds is given byspace v space equals space 4 t space minus space t squared space space for space 0 space less or equal than space space t space space less or equal than space 5

(i)
Explain why the particle is instantaneously at rest when t space equals space 0 space and space t space equals space 4. 

(ii)
Sketch a velocity-time graph for the motion of the particle during the interval 0 space less or equal than space space t space space less or equal than space 5.
7b
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4 marks

Use integration to show that 

(i)
the particle travels a distance of  32 over 3m between t space equals space 0 space and space t space equals space 4
(ii)
the particle travels a distance of   7 over 3m between t space equals space 4 space and space t space equals space 5.
7c
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3 marks

Use your answers to part (b) to 

(i)
find the total distance travelled by the particle between t = 0 and t = 5.

 

(ii)
Explain why the distance between the position of the particle at t = 0 and the position of the particle at t equals 5 is  25 over 3m.

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1a
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3 marks

A car is travelling along a straight horizontal motorway and passes a junction at time t space equals space 0 seconds. The car’s displacement, s metres, from the junction is then modelled by the equation

space space s space equals space 18 t squared space minus space t cubed

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

(ii)
Find the time, other than at t space equals space 0, that the model shows the car passing the same junction.
1b
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3 marks
(i)
Find an expression for the velocity, v space ms to the power of negative 1 end exponent comma of the car at time t seconds.
(ii)
Find the time, other than at t space equals space 0, that the model shows the car is instantaneously stationary.

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2a
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3 marks

A particle moving along a straight line has velocity v space straight m space straight s to the power of negative 1 end exponent, at time t seconds, and its motion is described the equation

v = t24t + 4           

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

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

2b
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2 marks

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

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3a
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2 marks

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

 s = 0.4t2 + 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.

3b
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3 marks

Show that the acceleration of the athlete should be constant, if they are to sprint the 100 m according to the model.

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4a
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5 marks

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 = 1 over 10t open parentheses 36 space minus space t close parentheses       space t space greater or equal than space 0

where v space straight m space straight s to the power of negative 1 end exponent 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.

4b
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2 marks

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

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5a
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3 marks

A home-made rocket is launched from rest at ground level with time t equals 0 space seconds.

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

 a equals 40 plus 6 t minus t squared                        t space greater or equal than space 0

(i)
Write down the acceleration of the rocket at launch.

(ii)
Find the acceleration of the rocket after 9 seconds.
5b
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4 marks
(i)
Find an expression for the velocity of the rocket at time t.

(ii)
Find an expression for the displacement of the rocket at time  t.

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6a
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4 marks

A particle moving along a horizontal path has acceleration a space m space s to the power of negative 2 end exponent at time t seconds modelled by the equation

a equals 13 minus 4 t space space space space space space space space space space space space space space space space t greater or equal than 0 

The particle has a velocity of 42 space m space s to the power of negative 1 end exponentat time t equals 2. Find an expression for the velocity of the particle at time t space seconds.

6b
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2 marks
(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.

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7a
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3 marks

In a cheese-rolling competition, a cylindrical block of cheese is rolled down a hill and its acceleration,a space m space s to the power of negative 2 end exponent , is modelled by the equation.

 a equals 1 plus 0.1 t                        0 less or equal than t less or equal than 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.

7b
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3 marks

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

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8a
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1 mark

A high-speed train has a maximum acceleration of 0.6 space m space s to the power of negative 2 end exponent which, from rest, takes 20 seconds to reach.

One such train leaves a station at t equals 0 seconds and its displacement, s space straight m, from the station is modelled using the equation

 s space equals space 1 over m t cubed space space space space space space space space space space space space space space space 0 less or equal than t less or equal than 20

 where m is a constant.

 

Find an expression for the velocity of the high-speed train for 0 space less or equal than t space less or equal than space 20.

8b
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3 marks
(i)
Find an expression for the acceleration of the high-speed train for 0 less or equal than t less or equal than 20.

(ii)
Thus find the value of the constant m, assuming that the train reaches its maximum acceleration in the quickest time possible.
8c
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2 marks

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

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9a
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3 marks

The velocity, v m s−1 , of a particle moving in a straight line at time t seconds is given by v equals 12 t minus 2 t ²  for 0 space less or equal than space space t space space less or equal than space space 10.

Sketch a velocity-time graph for the motion of the particle during the interval 0 space less or equal than space space t space space less or equal than space space 10.

9b
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5 marks

Use integration to show that the total distance travelled by the particle is 632 over 3 straight m.

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1a
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2 marks

A car is travelling along a straight horizontal motorway and passes under a bridge at time t space equals space 0 seconds. The car’s displacement, s metres, from the bridge is then modelled by the equation 

s equals t cubed minus 6 t squared

(i)
Find the displacement of the car from the bridge after 5 seconds.

(ii)
Find the time at which the model indicates the car passes under the bridge again.
1b
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3 marks
(i)
Find an expression for the velocity, v space straight m space straight s to the power of negative 1 end exponent comma of the car at time t seconds. 

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

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2a
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2 marks

A particle moving along a straight line has velocity, v space straight m space straight s to the power of negative 1 end exponent, at timeseconds according to the equation 

v space equals space t squared space minus space 6 t space plus space 8

Find the times at which the particle is instantaneously stationary.

2b
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3 marks

Find the distance travelled by the particle during the time it has negative velocity.

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3a
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2 marks

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

s equals 1.8 t plus 0.75 t squared minus 0.02 t cubed

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.

3b
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4 marks

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

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4a
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4 marks

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 equals 1 over w t squared open parentheses 60 space minus space t close parentheses 

where v space m space s to the power of negative 1 end exponent 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 space m space s to the power of negative 1 end exponent, find the value of w and the time at which the go kart reaches its maximum speed.

4b
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3 marks
(i)
Find the maximum acceleration of the new go kart model.

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

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5a
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3 marks

A home-made rocket is launched from rest, at time t = 0 seconds, from ground level with an acceleration of 56 space straight m space straight s to the power of negative 2 end exponent. The rocket’s acceleration is then modelled by the equation.                          

a space equals space 56 space plus space t space minus space t squared                              t space greater or equal than space 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 space straight m space straight s to the power of negative 1 end exponent.
5b
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4 marks

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

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6a
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5 marks

A zip-wire running between two trees in a children’s park is modelled as a horizontal line. The velocity-time graph below shows the motion of a child on the zip-wire as it moves from one tree to the other.

dWdTK1xL_picture1

 

The graph has the equation v equals 5 √ t for 0 less or equal than t less or equal than 4, where v space straight m space straight s to the power of negative 1 end exponent is the velocity at time t seconds. 

(i)
Find the distance between the two trees.

(ii)
Find the distance between the child and the second tree when the zip-wire comes to rest.
6b
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2 marks

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

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7a
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4 marks

A bullet train has a maximum acceleration of 0.72 space straight m space straight s to the power of negative 2 end exponent.

One such train leaves a station at timespace t space equals space 0 space seconds and its displacement, s space straight m, from the station is modelled using the equation 

s space equals space 3 over 200 t cubed space space space space space space space space space space space space space space space 0 space less or equal than space t space less or equal than space 8

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

7b
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3 marks

After reaching its maximum acceleration the bullet train continues to accelerate at that rate until its velocity reaches its maximum of 75 space straight m space straight s to the power of negative 1 end exponent.

How long does it take for this increase in velocity to happen?

7c
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3 marks

Once reaching its maximum velocity, the bullet train continues at this velocity for 10 minutes. Find the displacement of the train from the station at this time, giving your answer in kilometres to 3 significant figures.

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8a
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4 marks

The acceleration, a space straight m space straight s to the power of negative 2 end exponent, of a particle moving in a straight line at time straight t space seconds is given by a equals 4 t minus 7 for  0 less or equal than t less or equal than 6. Initially the velocity of the particle is 3 space straight m space straight s to the power of negative 1 end exponent.

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

8b
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5 marks

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

Show your method clearly.

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1a
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2 marks

A car is travelling along a straight horizontal motorway and passes a service station at time space t space equals space 0 spaceseconds. The car’s displacement, s metres, from the service station is then modelled by the equation

s space equals space 0.4 t space left parenthesis 2 t squared space minus space 4 t space plus space 3 right parenthesis

Show that the model indicates that the car never returns to the service station it passes at t space equals space 0 seconds.

1b
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4 marks

Show that the car is decelerating for the first 2 over 3 seconds after passing the service station.

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2a
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3 marks

A particle moving along a straight line has velocity,space v space straight m space straight s to the power of negative 1 end exponent, at time t seconds according to the equation

 v equals t cubed minus 12 t squared plus 39 t minus 28

Find the times at which the particle is instantaneously stationary.

2b
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3 marks

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

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3a
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4 marks

An athlete training for the 400 space straight m sprint is aiming to run according to the model

s equals 0.002 open parentheses 4000 t plus 50 t squared minus t cubed close parentheses

where  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.

(i)
Find the times that the athlete should pass the 100 m, 200 m and 300 m markers. Justify your choice of answers.
(ii)
Find the time the athlete should be aiming to finish the 400 m sprint in.
3b
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4 marks

Find the average acceleration, according to the model, for the last 50 m of the sprint. Comment on what the results indicate.

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4a
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3 marks

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 space equals space open curly brackets table row cell 0.25 left parenthesis t cubed minus 20 t squared plus 100 t right parenthesis space space space space space 0 less or equal than t less or equal than p end cell row cell 12 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space t greater or equal than p end cell end table close

where v space straight m space straight s to the power of negative 1 end exponent is the velocity of the go kart at time t seconds. p is an integer.

Find the value of p.

4b
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5 marks

Find the maximum and minimum velocities of the go kart in the first p seconds of its motion. Write down the acceleration of the go kart at these points.

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5
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7 marks

A home-made rocket is launched from rest at ground level at time t space equals space 0 seconds. Its acceleration is initially 64 space straight m space straight s to the power of negative 2 end exponent and is modelled by the equation

a space equals space 64 plus 12 t minus t squared space space space space space t greater or equal than 0

Find the total distance travelled by the rocket and the total time it spends in the air.

Give both answers to three significant figures and state any modelling assumptions you have made.

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6
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7 marks

In a cheese-rolling competition, a cylindrical block of cheese is rolled down a hill and its acceleration, a space straight m space straight s to the power of negative 2 end exponent , is modelled by the functions

 a left parenthesis t right parenthesis equals open curly brackets table row cell 0.2 space t space space space space space space space space space space 0 less or equal than t less or equal than 15 end cell row cell 9 minus t space space space space space space space space space space 15 less than t less or equal than A end cell end table close 

where t is the time in seconds and A is a constant. At the bottom of the hill the land is flat. The block of cheese comes to rest when its acceleration is negative 9 space straight m space straight s to the power of negative 2 end exponent.

By first finding the value of the constant A, find the distance the block of cheese rolls before it comes to rest.

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7
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8 marks

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

 s equals 1 over p space t to the power of q space space space space space space space space space space space end exponent 0 less or equal than t less or equal than 12

where p and q are constants.

In the first 10 seconds after the train leaves the station, the average velocity is  5 over 12 space straight m space straight s to the power of negative 1 end exponent and the average acceleration is  1 over 6 space straight m space straight s to the power of negative 2 end exponent.

By first finding the values of p and q, find an expression for the acceleration of the high-speed train for 0 space less or equal than space t space less or equal than space 12.

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8a
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5 marks

The acceleration, a space km space straight h to the power of negative 2 end exponent, of a particle moving in a straight line at time t hours is given by a equals 1 fifth space left parenthesis t minus 11 right parenthesis for  0 space less or equal than space t space less or equal than space 24. After 24 hours the particle has returned to where it started.

Show that the velocity, v space km space straight h to the power of negative 1 end exponent, of the particle at time t hours can be written as

 v equals 1 over 10 space left parenthesis t squared minus 22 t plus k right parenthesis

where k is a constant to be found.

8b
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4 marks

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

Show your method clearly.

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