OCR A Level Maths: Mechanics

Topic Questions

2.5 Constant Acceleration - 2D (A Level only)

1
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3 marks

A particle moves from rest and 8 seconds later has velocity open parentheses 3 bold i space plus 7 bold j close parentheses space straight m space straight s to the power of negative 1 end exponent.Find the displacement of the particle.

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2a
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1 mark

A ball is thrown from the top of a tall building with a velocity of open parentheses 2 bold i space plus space 29.4 bold j close parentheses space straight m space straight s to the power of negative 1 end exponent.

Write down the vector bold g space straight m space straight s to the power of negative 2 end exponent - the vector for acceleration due to gravity.

2b
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3 marks

How long does it take for the ball to reach a velocity of open parentheses 2 bold i space plus space 4.9 bold j close parentheses space straight m space straight s to the power of negative 1 end exponent?

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3a
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3 marks

A particle travels open parentheses negative 6 bold i space plus space 6 bold j close parentheses space straight m in 12 seconds with constant acceleration open parentheses 2 bold i space plus space bold j close parentheses space straight m space straight s to the power of negative 2 end exponent.

Find the velocity of the particle at the end of this motion.

 

3b
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2 marks

Use Pythagoras’ theorem to find the speed of the particle at the end of this motion, giving your answer to three significant figures.

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4a
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3 marks

A particle passes point A with velocity open parentheses 8 bold i space minus space 3 bold j close parentheses straight m space straight s to the power of negative 1 end exponent and 12 seconds later passes point B with velocity open parentheses negative 4 bold i space plus space 18 bold j close parentheses space straight m space straight s to the power of negative 1 end exponent.

Given that it is constant, find the acceleration of the particle between the points A and B.

4b
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2 marks

Use Pythagoras’ theorem to find the magnitude of the acceleration of the particle between the points A and B, giving your answer to three significant figures.

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5
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3 marks

A particle moves with acceleration open parentheses table row cell negative 3 end cell row 4 end table close parentheses space straight m space straight s to the power of negative 2 end exponent and after 7 seconds of motion has velocity open parentheses table row 5 row 3 end table close parentheses space straight m space straight s to the power of negative 1 end exponent. Find the displacement of the particle in this time.

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6a
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1 mark

A ball is projected from the top of a tall building. 9 seconds later it has displacement open parentheses table row 6 row cell negative 8 end cell end table close parentheses space straight m from its starting position.

Use Pythagoras’ theorem to find the distance the ball is from its starting point after 9 seconds.

6b
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3 marks

Find the velocity with which the ball is projected.

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7
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3 marks

A particle passes a fixed point, O, with velocity open parentheses table row cell 2.3 end cell row cell 1.8 end cell end table close parentheses space straight m space straight s to the power of negative 1 end exponent and accelerates at a constant open parentheses table row cell 0.3 end cell row cell negative 0.1 end cell end table close parentheses space straight m space straight s to the power of negative 2 end exponent. Find the displacement of the particle from O 4.8 seconds later.

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8a
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3 marks

In two minutes, a particle travels open parentheses table row 1200 row 2400 end table close parentheses space straight m. It’s velocity at this point is open parentheses table row 40 row 60 end table close parentheses space straight m space straight s to the power of negative 1 end exponent.

Assuming it is constant during this motion, find the acceleration of the particle.

8b
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2 marks

Use Pythagoras’ theorem to find the magnitude of the acceleration.

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9
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3 marks

A particle travels open parentheses 10.6 bold i bold space minus space 21.2 bold j close parentheses space straight m in 10.6 seconds, at which point it has velocity open parentheses 2 bold i space minus space 4 bold j close parentheses space straight m space straight s to the power of negative 1 end exponent. Show that the particle was initially at rest.

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10a
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3 marks

A particle is projected from ground level such that after 1.8 seconds its displacement is open parentheses 2.7 bold i space plus space 3.6 bold j close parentheses space straight m.

Find the velocity of the particle after 1.8 seconds.

10b
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2 marks

Use Pythagoras’ theorem to find the speed of the particle after 1.8 seconds, giving your answer to three significant figures.

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11
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4 marks

A particle is travelling with constant acceleration open parentheses table row 3 row cell negative 2 end cell end table close parentheses space straight m space straight s to the power of negative 2 end exponent. After it has been displaced by open parentheses table row cell negative 42 end cell row 6 end table close parentheses space straight m the particle has velocity open parentheses table row 2 row cell negative 5 end cell end table close parentheses space straight m space straight s to the power of negative 1 end exponent. Find the time it takes for this motion to occur.

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1a
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4 marks

Starting from rest a toy boat sails across a pond such that for the first 10 space seconds of its motion it has constant acceleration open parentheses 0.1 bold i space plus space 0.3 bold j close parentheses space straight m space straight s to the power of negative 2 end exponent. 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.

1b
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1 mark

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

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2a
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4 marks

A ball is thrown from the top of a tall building with a velocity of open parentheses table row cell 3.5 end cell row 6 end table close parentheses space straight m space straight s to the power of negative 1 end exponent.

(i)
Find the velocity of the ball 3 seconds after it is thrown.

(ii)
Find the speed of the ball 3 seconds after it is thrown.
2b
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3 marks

The ball first strikes the ground after 6 seconds.
Find the displacement of the ball when it first strikes the ground.

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3a
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3 marks

A particle travels left parenthesis 16 bold i bold space plus space 96 bold j right parenthesis space straight m in 8 seconds with a constant acceleration of left parenthesis a bold i bold space minus space 4 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent.

Given that the particle’s velocity after the 8 seconds is left parenthesis 14 bold i space plus space straight b bold j right parenthesis space straight m space straight s to the power of negative 1 end exponent find the values of the constants a and b.

3b
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2 marks

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

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4a
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5 marks

Two stones are slid across a large icy pond. The first stone is released from rest at the origin with constant acceleration left parenthesis 2 bold i space plus space 3 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent. The second stone is released from rest from the position with coordinates left parenthesis 50 comma space minus 100 right parenthesis with constant acceleration  left parenthesis bold i space plus space 5 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent.

 

(i)
Find the position vectors of both stones after 5 seconds.

(ii)
Find the distance between the two stones after 5 seconds.
4b
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2 marks

Show that the two stones collide after 10 seconds.

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5a
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2 marks

A horse running across a large area of open countryside starts to gallop with constant acceleration (0.6i + 0.4j) m s−2. After 12 seconds of galloping the horse has velocity (8i + 10j) m s−1.

Find the displacement of the horse at the end of its 12 second gallop.

5b
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4 marks

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

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6
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6 marks

It takes 6 minutes for a particle to travel  open parentheses table row cell 5.94 end cell row cell 13.86 end cell end table close parentheses km with constant acceleration. The particle’s velocity at the start of the 6 minutes is one-tenth of its velocity at the end. Find the acceleration of the particle.

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7a
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4 marks

A football is kicked from the top of a hill and its motion modelled as moving in a 2D plane under the force of gravity only. The ball is kicked such that its initial velocity is left parenthesis 15 bold i plus 24 bold j right parenthesis space straight m space straight s to the power of negative 1 end exponent.

 

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

(ii)
Find the horizontal ground covered by the football at the time it first hits ground level.
7b
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3 marks

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

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8a
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3 marks

A particle moves under constant acceleration open parentheses table row cell 2 q space plus space 1 end cell row cell q minus 1 end cell end table close parentheses space straight m space straight s to the power of negative 2 end exponent. In a 6 second time period, the particle has initial velocity open parentheses table row cell 5.2 end cell row cell 1 minus 2 p end cell end table close parentheses space straight m space straight s to the power of negative 1 end exponentand final velocity open parentheses table row cell 27 p plus 4 end cell row cell 5.2 end cell end table close parentheses space straight m space straight s to the power of negative 1 end exponent. p and q are constants.

 

Show that


27 p minus 12 q equals 7.2 and 

2 p minus 6 q equals negative 10.2
     

8b
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2 marks

Find the values of p and q.

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9a
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4 marks

A train leaves station O from rest with constant acceleration bold italic a equals left parenthesis 0.5 bold i space plus space 0.2 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent.
60 seconds later it passes (but does not stop at) station A at which point its acceleration changes to bold italic a equals left parenthesis 0.8 bold i space plus space 0.1 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent. 90 seconds later the train passes through station B.

(i)
Find the displacement of the train between station O and station B.

(ii)
Find the distance between station O and station B.
9b
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3 marks

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

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1
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4 marks

Starting from rest a toy boat experiencing a constant acceleration of open parentheses 0.5 bold i space plus space 0.2 bold j close parentheses space straight m space straight s to the power of negative 2 end exponent takes 12 seconds to sail across a pond.  Find the distance the toy boat sails across the pond, giving your answer to three significant figures.

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2a
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3 marks

A ball is thrown from the top of a tall building with a velocity of  open parentheses table row cell 0.8 end cell row cell 0.2 end cell end table close parentheses space straight m space straight s to the power of negative 1 end exponent.

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

2b
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2 marks

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

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3a
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3 marks

A particle experiences a constant acceleration of open parentheses 3 straight p bold i space plus space 2 bold j close parentheses space straight m space straight s to the power of negative 2 end exponent where p is a constant.

In 7 seconds the particle travels left parenthesis negative 91 bold i plus 7 bold j right parenthesis space space straight m.

Given that the particle’s velocity after the 7 seconds is open parentheses 4 straight p bold i space plus space 8 bold j close parentheses space straight m space straight s to the power of negative 1 end exponent find the value of the constant p.

3b
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2 marks

Find the magnitude of the acceleration during this motion.

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4a
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5 marks

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 open parentheses 3 bold i space minus bold space bold j close parentheses space straight m space straight s to the power of negative 2 end exponent.  The second stone experiences a constant acceleration of open parentheses negative 2 bold i space plus space bold j close parentheses space straight m space straight s to the power of negative 2 end exponent

(i)
Find the displacement of both stones after 8 seconds.

(ii)
Find the distance between the two stones after 8 seconds.
4b
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2 marks

If the second stone had been released from the point with coordinates open parentheses 10 space comma space 30 close parentheses rather than the origin, what would its position vector (rather than displacement from starting point) be at time t?

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5a
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2 marks

A horse running across a large area of open countryside starts to gallop with constant acceleration open parentheses 0.25 bold i space plus space 0.45 bold j close parentheses space straight m space straight s to the power of negative 2 end exponent. After 16 seconds of galloping the horse has velocity open parentheses 12 bold i space plus space 6 bold j close parentheses space straight m space straight s to the power of negative 1 end exponent.

Find the displacement of the horse from its starting position at the end of the 16 second gallop.

5b
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3 marks

Find the average velocity of the horse during the gallop.

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6
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4 marks

It takes four minutes for a particle to travel open parentheses table row cell 0.96 end cell row cell 1.2 end cell end table close parentheses 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.

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7a
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3 marks

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 open parentheses 18 bold i space plus space 23 bold j close parentheses space straight m space straight s to the power of negative 1 end exponent and it lands at ground level with velocity open parentheses 18 bold i space minus space 26 bold j close parentheses space straight m space straight s to the power of negative 1 end exponent, 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.

7b
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3 marks
(i)
Find the displacement of the football when it first hits ground level.

(ii)
Give an interpretation of the values of the components of the vector from part (b) (i).

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8a
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2 marks

A particle moves under constant acceleration.  In a 9 second time period the particle has initial velocity open parentheses table row cell 2 straight q space minus space 1 end cell row straight q end table close parentheses space straight m space straight s to the power of negative 1 end exponent and final velocity open parentheses table row straight p row cell 1 space minus space 8 straight p end cell end table close parentheses space straight m space straight s to the power of negative 1 end exponent,  where p and q are constants.  The displacement of the particle in these 9 seconds is open parentheses table row cell negative 22.5 end cell row cell negative 81 end cell end table close parentheses space straight m.

Show that

open parentheses table row cell negative 22.5 end cell row cell negative 81 end cell end table close parentheses space equals space 4.5 space open parentheses table row cell straight p plus 2 straight q minus 1 end cell row cell 1 minus 8 straight p plus straight q end cell end table close parentheses

8b
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3 marks

Show that the two simultaneous equations


p space plus space 2 q space equals space minus 4 space space space space space space space space space space a n d space space space space space space space space space space 8 p space minus space q space equals 19

must be satisfied and hence find the values of p and q.

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9a
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2 marks

A train leaves station O from rest with constant acceleration bold a space equals space left parenthesis 0.3 bold i space plus space 0.7 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent.
80 seconds later it passes through (but does not stop at) station A at which point its acceleration changes to bold a space equals space left parenthesis 0.5 bold i space plus space 0.3 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent. 180 seconds later the train passes through station B.

 

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

9b
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2 marks

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

9c
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2 marks

Find the displacement of the train between stations A and B.

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1
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5 marks

Starting from rest a toy boat sails across a pond such that for the first 15 seconds of its motion it has constant acceleration left parenthesis 0.12 bold i space plus space 0.05 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent. 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.

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2a
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3 marks

A ball is thrown from the top of a tall building with velocity  open parentheses table row 4 row cell 8.5 end cell end table close parentheses space straight m space straight s to the power of negative 1 end exponent.

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

2b
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5 marks
(i)
The ball first strikes the ground after 5 seconds. Find the height of the building and the distance from the building the ball is when it first strikes the ground.

(ii)
Find the distance between the point where the ball first strikes the ground and its starting point at the top of the building.

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3
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5 marks

A particle travels left parenthesis 464 bold i space minus space 272 bold j right parenthesis space straight m in 16 seconds with constant acceleration open parentheses negative p over q bold i plus fraction numerator 2 p over denominator q end fraction bold j close parentheses space straight m space straight s to the power of negative 2 end exponent. p and q are positive constants.

Given that the particle’s velocity after the 16 seconds is left parenthesis left parenthesis 2 q plus 3 right parenthesis bold i plus left parenthesis p plus 5 right parenthesis bold j right parenthesis space straight m space straight s to the power of negative 1 end exponent find the values of p and q.

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4
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7 marks

Two stones are slid across a large icy pond.

At time t space equals space 0 seconds, the first stone is located at the point with coordinates left parenthesis 2 space comma space minus 8 right parenthesis comma has initial velocity left parenthesis 5.4 bold i bold space plus space 7.2 bold j right parenthesis space straight m space straight s to the power of negative 1 end exponent and moves with constant acceleration left parenthesis 0.4 bold i bold space plus space 0.6 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent.

At time t space equals space 0 seconds, the second stone is located at the point with coordinates left parenthesis 50 space comma space 40 right parenthesis, has initial velocity left parenthesis negative 4.6 bold i space minus space 2.8 bold j right parenthesis space straight m space straight s to the power of negative 1 end exponent and moves with constant acceleration left parenthesis 1.4 bold i space plus space 1.6 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent.

 

Determine the two times at which the stones collide, and their distance from the origin at these times. Suggest a reason why the stones may not collide at the later time.

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5a
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2 marks

A horse running across a large area of open countryside starts to gallop with constant acceleration left parenthesis 0.3 bold i space plus space 0.2 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent. After 9 seconds of galloping the horse has velocity left parenthesis 10 bold i space minus space 7 bold j right parenthesis space straight m space straight s to the power of negative 1 end exponent.

Find the displacement of the horse at the end of the 9 second gallop.

5b
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4 marks
(i)
Find the average velocity of the horse during the gallop.
(ii)
Find the average speed of the horse during the gallop.
(iii)
Write down the magnitude of the average velocity and justify why you can do this without making any further calculations.

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6
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6 marks

It takes three minutes for a particle to travel open parentheses table row cell 1.62 end cell row cell 2.16 end cell end table close parentheses space space 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 magnitude of the acceleration giving your answer as a fraction, in metres per square second.

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7a
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3 marks

A football is kicked from the top of a hill and its motion modelled as moving in a 2D plane under gravity. Its initial velocity is left parenthesis 12 bold i space plus space 7 bold j right parenthesis space straight m space straight s to the power of negative 1 end exponent and it first hits ground level with velocity left parenthesis 12 bold i minus space 20 bold j right parenthesis space straight m space straight s to the power of negative 1 end exponent.

Find the height of the hill at the point from which the football was kicked.

7b
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4 marks

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

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8
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6 marks

A particle passes a fixed point O at time t = 0 seconds. T seconds later the particle has velocity open parentheses table row cell 2 T space plus space 4 end cell row cell 1 minus T end cell end table close parentheses space straight m space straight s to the power of negative 1 end exponent. The acceleration of the particle is constant throughout this motion at open parentheses table row cell T space minus space 3 end cell row cell T minus 9 end cell end table close parentheses space straight m space straight s to the power of negative 2 end exponent. The displacement of the particle from O T seconds after it passes O is open parentheses table row cell 4 T end cell row T end table close parentheses space straight m.  Show that the initial velocity is open parentheses table row cell negative 10 end cell row 8 end table close parentheses space straight m space straight s to the power of negative 1 end exponent.

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9
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7 marks

A train leaves station O from rest with constant acceleration bold a bold space equals space left parenthesis 0.4 bold i space plus space 0.1 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent.
2.5 minutes later it passes (but does not stop at) station A at which point its acceleration changes to bold a space equals space left parenthesis 0.2 bold i space minus space 0.3 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent. 5 minutes later the train passes through station B.

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

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