Newton’s laws of motion

A car of mass 1000 kg is moving along a straight horizontal road. The engine of the car produces a constant driving force of 800 N. At time t seconds, t 0, the speed of the car is vm s^–1 and the resistance to the motion of the car has magnitude 2v²N.

The time taken for the speed of the car to increase from 5 m s^–1 to 15 m s^–1 is T seconds.

When t = 0, v = 2

Show that pe^qt where p and q are rational numbers to be found.

Hence explain why v 20 for all values of t 0

A particle of mass 2 kg is moving in a straight line on a smooth horizontal surface under the action of a horizontal force of magnitude F newtons.

At time t seconds (t 0),

• the particle is moving with speed v ms¹

• F 2 v

The time taken for the speed of the particle to increase from 5ms¹ to 10ms¹ is T seconds.

The distance moved by the particle as its speed increases from 5ms¹ to 10ms¹ is D metres.

Find the exact value of D.

A cyclist and her cycle have a combined mass of 60 kg. The cyclist is moving along a straight horizontal road and is working at a constant rate of 200 W.

When she has travelled a distance x metres, her speed is vm s^–1 and the magnitude of the resistance to motion is 3v²N.

The distance travelled by the cyclist as her speed increases from 2 m s^–1 to 4 m s^–1 is D metres.

Find the exact value of D

At time t = 0, a small stone P of mass m is released from rest and falls vertically through the air. At time t, the speed of P is v and the resistance to the motion of P from the air is modelled as a force of magnitude kv², where k is a constant.

Give an interpretation of the value of V, justifying your answer.

At time t, P has fallen a distance s.

A particle P of mass 0.5 kg is moving along the positive x-axis in the direction of x increasing. At time t seconds (t 0), P is x metres from the origin O and the speed of P is vm s⁻¹. The resultant force acting on P is directed towards O and has magnitude kv²N, where k is a positive constant.

When x= 1, v= 4 and when x= 2, v= 2

Show that v=ab^x, where a and b are constants to be found.

The time taken for the speed of P to decrease from 4 m s⁻¹ to 2 m s⁻¹ is T seconds.