Calculus in kinematics

A rambler starts a walk at the bottom of a hill at 10 am.

The rambler walks all the way to the top of the hill and then turns around and walks back down to the bottom of the hill.

The differential equation

where A is a constant, is used to model the vertical displacement, x metres, of the rambler from the bottom of the hill, t hours after the start of the walk.

Given that after 1 hour

• the rambler has a vertical displacement of 213 m

determine the value of A to 3 significant figures.

Hence, determine the particular solution of the differential equation, giving your answer in the form x = f (t)

Use the model to determine the time at which the rambler will return to the bottom of the hill.

Given that the vertical displacement of the top of the hill is 300.68 m

use the model to find the value of t when the rambler reaches the top of the hill.

Using your answers to parts (c) and (d), give a limitation of the model.

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 raindrop falls from rest from a cloud. The velocity, vm s^–1 vertically downwards, of the raindrop, t seconds after the raindrop starts to fall, is modelled by the differential equation

where k is a positive constant.

Solve the differential equation to show that

Given that v = 4 whent = 2

determine, according to the model, the velocity of the raindrop 5 seconds after it starts to fall.

Comment on the validity of the model for very large values of t

A raindrop falls from rest from a cloud. The velocity, vm s^–1 vertically downwards, of the raindrop, t seconds after the raindrop starts to fall, is modelled by the differential equation

where k is a positive constant.

Solve the differential equation to show that

Given that v = 4 whent = 2

determine, according to the model, the velocity of the raindrop 5 seconds after it starts to fall.

Comment on the validity of the model for very large values of t

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A particle P moves along a straight line. Initially P is at rest at the point O on the line.

At time t seconds, where t 0

• the displacement of P from O is x metres

• the velocity of P is vm s^–1 in the positive x direction

Show that, at time t seconds, v p , where p and q are constants to be determined.

Find the limiting value of v as t increases.

Find the value of x whent = 2

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.

The motion of a pendulum, shown in Figure 3, is modelled by the differential equation

where θ is the angle, in radians, that the pendulum makes with the downward vertical, t seconds after it begins to move.

Show that a particular solution of the differential equation is

Hence, find the general solution of the differential equation.

Initially, the pendulum

• makes an angle of radians with the downward vertical

• is at rest

Given that, 10 seconds after it begins to move, the pendulum makes an angle of α radians with the downward vertical,

determine, according to the model, the value of α to 3 significant figures.

Given that the true value of α is 0.62

evaluate the model.

The differential equation

models the motion of the pendulum as moving with forced harmonic motion.

Refine the differential equation so that the motion of the pendulum is simple harmonic motion.

The motion of a pendulum, shown in Figure 3, is modelled by the differential equation

where θ is the angle, in radians, that the pendulum makes with the downward vertical, t seconds after it begins to move.

Show that a particular solution of the differential equation is

Hence, find the general solution of the differential equation.

Initially, the pendulum

• makes an angle of radians with the downward vertical

• is at rest

Given that, 10 seconds after it begins to move, the pendulum makes an angle of α radians with the downward vertical,

determine, according to the model, the value of α to 3 significant figures.

Given that the true value of α is 0.62

evaluate the model.

The differential equation

models the motion of the pendulum as moving with forced harmonic motion.

Refine the differential equation so that the motion of the pendulum is simple harmonic motion.

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.

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.