Resisted motion

A particle P of mass m is fired vertically upwards from a point on the surface of the Earth. Initially P moves in a straight line directly away from the centre of the Earth. When P is at a distance x from the centre of the Earth, the gravitational force exerted by the Earth on P is directed towards the centre of the Earth and has magnitude , where k is a constant. At the surface of the Earth the acceleration due to gravity is g. The Earth is modelled as a fixed sphere of radius R.

Show that k = mgR².

When P is at a height above the surface of the Earth, P is moving away from

find, in terms of R, the greatest height of P above the surface of the Earth.

A particle of mass m is projected vertically upwards in a resisting medium. As the particle moves upwards, the speed v of the particle is given by

where x is the distance of the particle above the point of projection and k is a positive constant.

Show that the magnitude of the resistance to the motion of the particle is

Find, in terms of k, the greatest height reached by the particle above the point of projection.

Show that the time taken by the particle to reach its greatest height above the point of

A particle P of mass m kg is initially held at rest at the point O on a smooth plane which is inclined at 30° to the horizontal. The particle is released from rest and slides down the plane against a force of magnitude mx newtons acting towards O, where x metres is the distance of P from O.

Find the speed of P when x = 3

Find the distance P has moved when it first comes to instantaneous rest.

A cyclist and her bicycle have a combined mass of 75 kg. The cyclist travels along a straight horizontal road. The cyclist produces a constant driving force agnitude

cyclist moves, the total resistance to motion of the cyclist and her bicycle has magnitude 3v² newtons. The cyclist starts from rest. At time t seconds, she has travelled a distance x metres from her starting point. Find

v in terms of x,

t in terms of v.

One end of a light elastic string, of natural length 1.5m and modulus of elasticity 14.7N, is attached to a fixed point O on a ceiling. A particle P of mass 0.6kg is attached to the free end of the string. The particle is held at O and released from rest. The particle comes to instantaneous rest for the first time at the point A. Find

the distance OA,

the magnitude of the instantaneous acceleration of P at A.

A vehicle of mass 900 kg moves along a straight horizontal road. At time t seconds the resultant force acting on the vehicle has magnitude N, where k is a positive constant. The force acts in the direction of motion of the vehicle. At time t seconds, t 1, the speed of the vehicle is v m s^–1 and the vehicle is a distance x metres from a fixed point O on the road. When t = 1 the vehicle is at rest at O and when t = 4 the speed of the vehicle is 10.5 m s^–1.

Hence deduce that the speed of the vehicle never reaches 14 m s^–1.

Use the trapezium rule, with 4 equal intervals, to estimate the value of x when v = 7

A car of mass 900 kg is moving along a straight horizontal road with the engine of the car working at a constant rate of 22.5 kW. At time t seconds, the speed of the car is v m s(0 v 30) and the total resistance to the motion of the car has magnitude 25 v newtons.

Show that when the speed of the car is v m s^í1, the acceleration of the car is

The time taken for the car to accelerate from 10 m s^í1 to 20 m s^í1 is T seconds.

Show that

Find the distance travelled by the car as it accelerates from 10 m s to 20 m s

A car of mass 1000 kg is moving along a straight horizontal road. The engine of the car is working at a constant rate of 25 kW. When the speed of the car is v m s^–1, the resistance to motion has magnitude 10 v newtons.

Show that, at the instant when v = 20, the acceleration of the car is 1.05 m s^–2.

Find the distance travelled by the car as it accelerates from a speed of 10 m s^–1 to a speed of 20 m s^–1.

A small ball of mass m is projected vertically upwards from a point O with speed U. The ball is subject to air resistance of magnitude mkv, where v is the speed of the ball and k is a positive constant. Find, in terms of U, g and k, the maximum height above O reached by the ball.

A particle of mass 4 kg is moving along the horizontal x-axis under the action of a single force which acts in the positive x-direction. At time t seconds the force has magnitude

When t = 0 the particle has speed 2 m s^–1 in the positive x-direction. Find the work done by the force in the interval 0 ≤ t 4

A particle P of mass 0.5 kg falls vertically from rest. After t seconds it has speed v m s^–1. A resisting force of magnitude 1.5 v newtons acts on P as it falls.

Show that 3v = 9.8(1 – e^–3t).

Find the distance that P falls in the first two seconds of its motion.

A van of mass 1200 kg travels along a straight horizontal road against a resistance to motion which is proportional to the speed of the van. The engine of the van is working at a constant rate of 40 kW. The van starts from rest at time t = 0. At time t seconds, the speed of the van is v m s^–1. When the speed of the van is 40 m s^–1, the acceleration of the van is 0.3 m s^–2.

Show that

Find v in terms of t.

A particle P of mass 0.6 kg is moving along the x-axis in the positive direction. At time t = 0, P passes through the origin O with speed 15 m s^–1. At time t seconds the distance OP is x metres, the speed of P is v m s^–1 and the resultant force acting on P has magnitude newtons. The resultant force is directed towards O.

Find the value of x when t = 5

A particle P of mass 1.5 kg is attached to the mid-point of a light elastic string of natural length 0.30 m and modulus of elasticity newtons. The ends of the string are attached to two fixed points A and B, where AB is horizontal and AB = 0.48 m. Initially P is held at rest at the mid-point, M, of the line AB and the tension in the string is 240 N.

Show that = 400

The particle is now held at rest at the point C, where C is 0.07 m vertically below M. The particle is released from rest at C.

Find the magnitude of the initial acceleration of P.

Find the speed of P as it passes through M.

A particle P of mass 0.5 kg moves on the positive x-axis under the action of a single force directed towards the origin O. At time t seconds the distance of P from O is x metres, the magnitude of the force is 0.375x² N and the speed of P is v m s^–1. When t = 0, OP = 8 m and P is moving towards O with speed 2 m s^–1.

Find the distance of P from O at the instant when v = 5.

A particle Q of mass 6 kg is moving along the x-axis. At time t seconds the displacement of Q from the origin O is x metres and the speed of Q is v m s^–1. The particle moves under the action of a retarding force of magnitude (a + bv²) N, where a and b are positive constants. At time t = 0, Q is at O and moving with speed U m s^–1 in the positive x-direction. The particle Q comes to instantaneous rest at the point X.

Show that the distance OX is

Given that a = 12 and b = 3,

find, in terms of U, the time taken to move from O to X.

A particle P of mass m is above the surface of the Earth at distance x from the centre of the Earth. The Earth exerts a gravitational force on P. The magnitude of this force is inversely proportional to x². At the surface of the Earth the acceleration due to gravity is g. The Earth is modelled as a sphere of radius R.

A particle is fired vertically upwards from the surface of the Earth with initial speed 3 U. At a height R above the surface of the Earth the speed of the particle is U.

Find U in terms of g and R.

A particle of mass m is projected vertically upwards, at time t = 0, with speed U. The particle is subject to air resistance of magnitude , where v is the speed of the particle at time t and k is a positive constant.

Show that the particle reaches its greatest height above the point of projection at time

Find the greatest height above the point of projection attained by the particle.

A particle P of mass 0.5 kg is moving along the positive x-axis. At time t seconds, P is moving under the action of a single force of magnitude 4 cosπ t N, directed away from the origin. When t = 1, the particle P is moving away from the origin with speed 6 m s^–1. Find the speed of P when t = 1.5, giving your answer to 3 significant figures.

A light elastic string has natural length a and modulus of elasticity mg. A particle P of mass m is attached to one end of the string. The other end of the string is attached to a fixed point A. The particle is released from rest at A and falls vertically. When P has fallen a distance a + x, where x > 0, the speed of P is v.

Find the greatest speed attained by P as it falls.

After release, P next comes to instantaneous rest at a point D.

Find the magnitude of the acceleration of P at D.

A cyclist and her bicycle have a combined mass of 100 kg. She is working at a constant rate of 80 W and is moving in a straight line on a horizontal road. The resistance to motion is proportional to the square of her speed. Her initial speed is 4 m s^–1 and her maximum possible speed under these conditions is 20 m s^–1. When she is at a distance x m from a fixed point O on the road, she is moving with speed v m s^–1 away from O.

Show that

Find the distance she travels as her speed increases from 4 m s^–1 to 8 m s^–1.

Use the trapezium rule, with 2 intervals, to estimate how long it takes for her speed to increase from 4 m s^–1 to 8 m s^–1.

k is a constant. At time t the speed of P is v. The particle P moves against air resistance whose magnitude is modelled as being mkv² when the speed of P is v. Find, in terms of k, the distance travelled by P until its speed first becomes half of its initial speed.

A particle P of mass 3 kg is moving in a straight line. At time t seconds, 0 t 4, the only force acting on P is a resistance to motion of magnitude 9+⎜9 + N. At ⎟ time t seconds the velocity of P is v m s^–1. When t = 4, v = 0. Find the value of v when t = 0.

A particle P of mass 0.5 kg moves along the positive x-axis. It moves away from the origin O under the action of a single force directed away from O. When OP = x metres, the magnitude of the force is N and the speed of P is v m s^–1. Initially P is at rest at O.

Show that the speed of P never reaches √6 m s^–1.

Find x when P has been moving for 2 seconds.

At time t = 0, a particle of mass m is projected vertically downwards with speed U from a point above the ground. At time t the speed of the particle is v and the magnitude of the air resistance is modelled as being mkv, where k is a constant. Given that U < , find, in terms of k, U and g, the time taken for the particle to double its speed.

A particle P of mass 0.1 kg moves in a straight line on a smooth horizontal table. When P is a distance x metres from a fixed point O on the line, it experiences a force of magnitude N away from O in the direction OP. Initially P is at a point 2 m from O and is moving towards O with speed 8 m s^–1. Find the distance of P from O when P first comes to rest.