Calculus in kinematics

The displacement, s metres, of a particle P, at time t seconds, is given by

Find an expression for the velocity of particle P after t seconds.

The acceleration, a m s^–2, of a particle Q, at time t seconds, is given by

a = 6 t i + 7 j

Particle Q has an initial velocity of 4j m s^–1

Particles P and Q are moving parallel to each other when t = 2

Find the value of q

A car has an initial velocity of 1 m s¹

The car is moving in a straight line.

where k is a constant.

When t 3 the car has a velocity of 10 m s¹

A particle is moving such that its position vector, r metres, at time t seconds, is given by

Fully justify your answer.

A ball is released from a great height so that it falls vertically downwards towards the surface of the Earth.

Using a simple model, Andy predicts that the velocity of the ball, exactly 2 seconds

Show how Andy has obtained his prediction.

t seconds after being released from rest is

Find an expression for v in terms of t.

Comment on the value of v for the two models as t becomes large.

At time t seconds a particle, P, has position vector r metres, with respect to a fixed origin, such that

3 2 2

Find the exact speed of P when t 2

Bella claims that the magnitude of acceleration of P will never be zero.

Determine whether Bella’s claim is correct.

Fully justify your answer.

An elite athlete runs in a straight line to complete a 100-metre race.

where t is the time, in seconds, after the starting pistol is fired.

Find the maximum value of v, giving your answer to one decimal place.

Fully justify your answer.

Find an expression for the distance run in terms of t.

The athlete’s actual time for this race is 9.8 seconds.

Comment on the accuracy of the model.

A driver is road-testing two minibuses, A and B, for a taxi company.

The performance of each minibus along a straight track is compared.

A flag is dropped to indicate the start of the test.

Each minibus starts from rest.

The acceleration in m s² of each minibus is modelled as a function of time, t seconds, after the flag is dropped:

Find the time taken for A to travel 100 metres.

Give your answer to four significant figures.

The company decides to buy the minibus which travels 100 metres in the shortest time.

Determine which minibus should be bought.

The models assume that both minibuses start moving immediately when t 0

In light of this, explain why the company may, in reality, make the wrong decision.