Set 3 (Edexcel A Level Maths: Mechanics): Exam Questions

An ice skater moves across an ice rink such that their position, at time t seconds relative to an origin, is given by 

(i)

Briefly explain how you can tell the ice skater’s motion did not start at the origin.

(ii)

Find the coordinates of the position of the ice skater after 40 seconds.

(iii)

Find the distance between the ice skater and the origin after 40 seconds.

(i)

Find an expression for the velocity of the ice skater at time t seconds.

(ii)

Find an expression for the acceleration of the ice skater at time t seconds.

A train leaves station O from rest with constant acceleration 80 seconds later it passes through (but does not stop at) station A at which point its acceleration changes to . 180 seconds later the train passes through station B.

Find the displacement of the train from station O when it passes through station A.

Find the velocity of the train as it passes through station A.

Find the displacement of the train between stations A and B.

A train leaves station O from rest with constant acceleration 80 seconds later it passes through (but does not stop at) station A at which point its acceleration changes to . 180 seconds later the train passes through station B.

Find the displacement of the train from station O when it passes through station A.

Find the velocity of the train as it passes through station A.

Find the displacement of the train between stations A and B.

Two particles A and B, of identical mass, are connected by means of a light inextensible string. Particle A is held motionless on a rough fixed plane inclined at 30° to the horizontal, and that plane is connected at its top to another rough fixed plane inclined at 70° to the horizontal.  The string passes over a smooth light pulley fixed at the top of the two planes so that B is hanging downwards in contact with the second plane.   This situation is shown in the diagram below:

The parts of the string between A and the pulley and between B and the pulley each lie along a line of greatest slope of the respective planes. The coefficient of friction between the particles and the planes is 0.15 in both cases.

The system is released from rest with the string taut, and with particle B a vertical distance of from the ground.  Particle B descends along the slope until it reaches the ground, at which point it immediately comes to rest.  Particle A continues to move up the slope until the forces of gravity and friction cause it to come momentarily to rest.

Find the total distance travelled by particle A between the time that the system is first released from rest and the time that particle A comes momentarily to rest again after B has reached the ground.

Two particles A and B, of identical mass, are connected by means of a light inextensible string. Particle A is held motionless on a rough fixed plane inclined at 30° to the horizontal, and that plane is connected at its top to another rough fixed plane inclined at 70° to the horizontal.  The string passes over a smooth light pulley fixed at the top of the two planes so that B is hanging downwards in contact with the second plane.   This situation is shown in the diagram below:

The parts of the string between A and the pulley and between B and the pulley each lie along a line of greatest slope of the respective planes. The coefficient of friction between the particles and the planes is 0.15 in both cases.

The system is released from rest with the string taut, and with particle B a vertical distance of from the ground.  Particle B descends along the slope until it reaches the ground, at which point it immediately comes to rest.  Particle A continues to move up the slope until the forces of gravity and friction cause it to come momentarily to rest.

Find the total distance travelled by particle A between the time that the system is first released from rest and the time that particle A comes momentarily to rest again after B has reached the ground.

A manager at the company Rods-We-Are has invented a device for locating the centre of mass of the 5 metre long barge poles that the company sells. He has connected force metres to two smooth supports located 2.5 m apart at the same horizontal level. A barge pole is placed on the supports so that it is held horizontally in equilibrium, with the supports located at points C and D as indicated in the diagram below:

The pole is then slid back and forth on the supports until a buzzer sounds, which indicates that the reaction force at C is exactly forty-nine times the reaction force at D. The distance x from the left end of the pole in the diagram to C is then measured.  Finally, by modelling the barge pole as a rod, the value of x is used to calculate the distance d between the left end of the pole in the diagram and the centre of mass of the pole.

A barge pole is placed on the device described above, and the buzzer sounds when . By first finding an expression for d in terms of x, determine the location of the centre of mass of the barge pole.

The manager claims that although he has seen many non-uniform 5 metre barge poles, he has never found one for which the centre of mass could not be determined using his device.  A new trainee manager claims that her calculations show that there could be 5 metre barge poles for which the device will not be able to determine the centre of mass.

Explain why both the manager and the trainee manager could be correct, supporting your answer with precise mathematical reasoning.

A manager at the company Rods-We-Are has invented a device for locating the centre of mass of the 5 metre long barge poles that the company sells. He has connected force metres to two smooth supports located 2.5 m apart at the same horizontal level. A barge pole is placed on the supports so that it is held horizontally in equilibrium, with the supports located at points C and D as indicated in the diagram below:

The pole is then slid back and forth on the supports until a buzzer sounds, which indicates that the reaction force at C is exactly forty-nine times the reaction force at D. The distance x from the left end of the pole in the diagram to C is then measured.  Finally, by modelling the barge pole as a rod, the value of x is used to calculate the distance d between the left end of the pole in the diagram and the centre of mass of the pole.

A barge pole is placed on the device described above, and the buzzer sounds when . By first finding an expression for d in terms of x, determine the location of the centre of mass of the barge pole.

The manager claims that although he has seen many non-uniform 5 metre barge poles, he has never found one for which the centre of mass could not be determined using his device.  A new trainee manager claims that her calculations show that there could be 5 metre barge poles for which the device will not be able to determine the centre of mass.

Explain why both the manager and the trainee manager could be correct, supporting your answer with precise mathematical reasoning.

In Toonland, a coyote is desperately trying to catch the very fast roadrunner bird.  In its latest effort to keep pace with the roadrunner the coyote projects itself from a catapult at the top of a canyon 85 m tall.  The catapult projects the coyote with initial velocity .

The roadrunner spots the coyote’s plan when the coyote is at its maximum height above the ground.  Using magic Toon paint the roadrunner paints a hole on the ground at the spot where the coyote will land.

It takes the roadrunner 4 seconds to paint the hole on the ground, and once it is finished the magic of the paint will cause it to become (at least for coyotes) a real hole with no bottom.  Determine whether or not the roadrunner will succeed in causing the coyote to plummet endlessly to its doom.

In Toonland, a coyote is desperately trying to catch the very fast roadrunner bird.  In its latest effort to keep pace with the roadrunner the coyote projects itself from a catapult at the top of a canyon 85 m tall.  The catapult projects the coyote with initial velocity .

The roadrunner spots the coyote’s plan when the coyote is at its maximum height above the ground.  Using magic Toon paint the roadrunner paints a hole on the ground at the spot where the coyote will land.

It takes the roadrunner 4 seconds to paint the hole on the ground, and once it is finished the magic of the paint will cause it to become (at least for coyotes) a real hole with no bottom.  Determine whether or not the roadrunner will succeed in causing the coyote to plummet endlessly to its doom.

In the annual Magical Beasts’ relay, each team competes over a 600  course. Teams consists of three magical beasts, one of the land, one of the sky and one of the sea. Each beast covers a distance of 200  moving at a constant velocity in a straight line. Last year the defending champions set a new record, covering the course in 4 hours 43 minutes and 17 seconds. This year the challenging team is made up of a Unicorn, a Dragon and a Mermaid. The Unicorn runs at a velocity of , the Dragon flies at a velocity of and finally, the Mermaid swims at a velocity of per minute.

Can this year’s challenging team break the course record?

Your final answer must include the time difference to the nearest second.

In the annual Magical Beasts’ relay, each team competes over a 600  course. Teams consists of three magical beasts, one of the land, one of the sky and one of the sea. Each beast covers a distance of 200  moving at a constant velocity in a straight line. Last year the defending champions set a new record, covering the course in 4 hours 43 minutes and 17 seconds. This year the challenging team is made up of a Unicorn, a Dragon and a Mermaid. The Unicorn runs at a velocity of , the Dragon flies at a velocity of and finally, the Mermaid swims at a velocity of per minute.

Can this year’s challenging team break the course record?

Your final answer must include the time difference to the nearest second.