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IB Maths AA SL 5.9 Kinematics

IB Maths AA SL 5.9 Kinematics
IB Mathematics: analysis and approaches guide, first assessment 2021

Practise linking displacement, velocity and acceleration with derivatives and integrals in particle, vehicle and race contexts.

How this is tested

  • use a=v'(t) or equal-speed equations to find times from velocity functions in motion models
  • set up ∫v(t) dt or ∫|v(t)| dt to distinguish displacement from total distance travelled
  • decide whether an object returns to its start by evaluating signed displacement

Question 9

[Maximum number: 8]

An airplane lands on a runway 100 metres in front of a stationary car. At the instant the airplane lands, the car begins to travel in the same direction towards the airplane.

diagram not to scale

diagram not to scale

Let t represent the number of seconds after the airplane lands. For t≥0t \geq 0, the velocities of the airplane and the car in ms−1\mathrm{ms}^{-1}, can be modelled by the following equations:

vair =60e−0.1tvcar =5t\begin{aligned} & v_{\text {air }}=60 \mathrm{e}^{-0.1 t} \\ & v_{\text {car }}=5 t \end{aligned}

Question 9(a)

(a)

When the airplane lands, write down the speed of

[ 2 ]

Question 9(a)(i)

(i)

the airplane;

Question 9(a)(ii)

(ii)

the car.

[ 2 ]

Question 9(b)

(b)

Find

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Question 9(b)(i)

(i)

the value of t when the airplane and the car have the same speed;

Question 9(b)(ii)

(ii)

the speed at this time.

Let d(t) represent the distance, in metres, between the car and the back of the airplane after t seconds.

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Question 9(e)

(c)

Find the distance travelled by the car when it reaches the back of the airplane.

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