O LevelAdditional MathsCambridgeTopic QuestionsCalculusCalculus for Kinematics

Calculus for Kinematics (Cambridge O Level Additional Maths)

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1
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In this question, all lengths are in metres and time, t, is in seconds.

q7ai-june-2021-cie-igcse-additional-maths

The diagram shows the displacement–time graph for a runner, for 0 space less-than or slanted equal to space t space less-than or slanted equal to space 40.

(i)
Find the distance the runner has travelled when t space equals space 40.
(ii)
On the axes, draw the corresponding velocity–time graph for the runner, for 0 space less-than or slanted equal to space t space less-than or slanted equal to space 40.
q7aii-june-2021-cie-igcse-additional-maths

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2a
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A particle travels in a straight line. As it passes through a fixed point O, the particle is travelling at a velocity of 3 space ms to the power of – 1 end exponent. The particle continues at this velocity for 60 seconds then decelerates at a constant rate for 15 seconds to a velocity of 1.6 space ms to the power of – 1 end exponent. The particle then decelerates again at a constant rate for 5 seconds to reach point A space, where it stops.

Sketch the velocity-time graph for this journey on the axes below.

q9-0606-s20-qp-21-additional-maths

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2b
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Find the distance between O and A.

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2c
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Find the deceleration in the last 5 seconds.

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3
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q4ai-0606-w20-qp-13-additional-maths

The diagram shows the x – t graph for a runner, where displacement, x, is measured in metres and time, t, is measured in seconds.

(i)
On the axes below, draw the v – t graph for the runner.

[3]


q4aii-0606-w20-qp-13-additional-maths

(ii)
Find the total distance covered by the runner in 125 s.

[1]

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4
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A particle P moves in a straight line such that its displacement, x space straight m, from a fixed point O at time t space straight s is given by  x equals 10 space sin space 2 t minus 5.

(i)
Find the speed of P when t space equals space straight pi.

[1]

(ii)
Find the value of t for which P is first at rest.

[2]

(iii)
Find the acceleration of P when it is first at rest.

[2]

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5a
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In this question, the units are metres and seconds.
A particle P is travelling in a straight line. Its acceleration, a, away from a fixed point O, at time t, is given by a space equals space open parentheses 3 t space plus space 2 close parentheses to the power of negative 1 third end exponent, where t space greater-than or slanted equal to space 0.
Whenspace t space equals space 2, P is travelling with a velocity of 8 and has a displacement of – space 4.8 from O.
Find an expression for the velocity of P at time t.

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5b
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Explain why P is never at rest.

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5c
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Find the displacement of P from O when t space equals 25 over 3 .

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6
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A particle, P, moves in a straight line such that its displacement from a fixed point at time t is s.
The acceleration of P is given by open parentheses 2 t space plus space 4 close parentheses to the power of negative 1 half end exponent comma space for space t greater than 0

(i)
Given that P has a velocity of 9 when t space equals space 6, find the velocity of P at time t.


(ii)
Given that s = 1 third when t space equals space 6, find the displacement of P at time t.

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7a
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At time t straight s, a particle travelling in a straight line has acceleration open parentheses 2 t plus 1 close parentheses to the power of negative 1 half end exponent space ms to the power of negative 2 end exponent. When t space equals space 0, the particle is 4 space straight m from a fixed point O and is travelling with velocity 8 space ms to the power of negative 1 end exponent away from O.

Find the velocity of the particle at time t space straight s.

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7b
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Find the displacement of the particle from O at time t space straight s.

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1
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q9b-0606-s20-qp-12-additional-maths

The diagram shows the velocity–time graph for a particle Q travelling in a straight line with velocity v space ms to the power of negative 1 end exponent at time t space straight s. The particle accelerates at 3.5 space ms to the power of negative 2 end exponent for the first 10 s of its motion and then travels at constant velocity, V space ms to the power of negative 1 end exponent, for 10 space straight s. The particle then decelerates at a constant rate and comes to rest. The distance travelled during the interval 20 space less or equal than space t space less or equal than space 25 space i s space 112.5 space straight m.

(i)
Find the value of V.

[1]

(ii)
Find the velocity of Q when t space equals space 25.

[3]

(iii)
Find the value of t when Q comes to rest.

[3]

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2
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q12a-0606-w20-qp-12-additional-maths

The diagram shows the velocity–time graph of a particle P that travels 2775 m in 90 s, reaching a final velocity of V space ms to the power of negative 1 end exponent.

(i)
Find the value of V.

[3]

(ii)
Write down the acceleration of P when t space equals space 40.

[1]

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3
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The displacement,space x space straight m, of a particle from a fixed point at timespace t space straight s space is given by x equals space 6 space cos open parentheses 3 t plus straight pi over 3 close parentheses.
Find the acceleration of the particle when t equals fraction numerator 2 straight pi over denominator 3 end fraction.

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4a
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A particle P moves in a straight line such that, t seconds after passing through a fixed point O, its acceleration, a space ms to the power of negative 2 end exponent, is given by a space equals negative 6. When t space equals space 0, the velocity of P is 18 space ms to the power of negative 1 end exponent.

Find the time at which P comes to instantaneous rest.

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4b
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Find the distance travelled by P in the 3rd second.

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5
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The acceleration, a space ms to the power of negative 2 end exponent, of a particle Q travelling in a straight line, is given by a space equals space 6 space cos space 2 t at time t space straight s. When space t equals 0 space the particle is at point O and is travelling with a velocity of 10 space ms to the power of negative 1 end exponent.

(i)
Find the velocity of Q at time t.

[3]

(ii)
Find the displacement of Q from O at time t.

[3]

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1a
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A particle moves in a straight line such that, t seconds after passing a fixed point O, its displacement from O is s space straight m, where s space equals space straight e to the power of 2 t end exponent space minus 10 space straight e to the power of t space minus 12 t plus 9.

Find expressions for the velocity and acceleration at time t.

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1b
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Find the time when the particle is instantaneously at rest.

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1c
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Find the acceleration at this time.

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