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Solved problems for the chapter:

Momentum and its Conservation

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Momentum

1. Impulse - momentum theorem

2. Law of conservation of momentum

2.1. Elastic and Inelastic collisions

2.1.1. Elastic Collision

2.1.2. Inelastic collision

2.1.3. Partially elastic collision

2.2. External and internal force impulses

2.3. Law of conservation of momentum

3. Forum

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Kinetic Energy for A-Levels

1. Translational kinetic energy

1.1. Work - Kinetic energy theorem

2. Rotational kinetic energy

2.1. Moment of inertia of a real body

2.2. Kinetic energy of a rolling body

3. Forum

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Angular Momentum

1. Moment of inertia

1.1. Moment of inertia of a point body

1.2. Moment of inertia of a rigid body

2. Angular momentum

2.1. Conservation of angular momentum

2.2. Law of conservation of angular momentum

3. Forum

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Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:momentum_and_its_conservation/4021/

Example 1.

Two balls with equal masses of 0.4 kg move at a speed of :

in the same direction,
in the opposite direction and
perpendicular to each other.

What is the total momentum in each case?

Example 2.

A body's speed is increased 4 times. By how much is its kinetic energy increased?

A fisherman on a boat is approaching the shore at a speed of . The weight of the boat is 200 kg and the weight of the fisherman is 70 kg. Just before the pier, the fisherman jumps ashore. At what speed must he jump out of the boat (according to a stationary observer onshore) for the boat to stop in front of the pier?

A yacht and a boat are initially at rest on the water. The captain of the yacht then jumps from the yacht into the boat at a speed of . The yacht and the boat are now moving in their own directions. Given that the mass of the yacht is 2000 kg, the mass of the boat is 500 kg and the mass of the captain is 70 kg, what are the speeds of the yacht and that of the boat?

An object is stationary on the ground when it suddenly splits into three parts. The first two parts have the same mass and move perpendicular to each other. Their velocities are and . The third part is three times the mass of the other two. At what speed and in which direction is it moving?

A hunter weighing 70 kg stands on ice when he fires a bullet. The bullet has a mass of 20 g and an initial velocity of . Due to the recoil of the gun, the hunter slides back. By how much does he slide back if the coefficient of friction between his footwear and ice is 0.2?

Two objects move towards each other along the same straight line.

After colliding, the two objects stick together and are stationary. Which statement must be correct?

The total momentum of the two objects before the collision is zero.
The two objects have the same speed before the collision.
The total kinetic energy of the two objects does not change during the collision.
The two objects have equal mass.

An object of mass travelling with speed collides with, and sticks to, an object of mass travelling in the same direction with speed .

What is the speed with which the two objects travel together in the original direction?

What is a statement of the principle of conservation of momentum?

The momentum of an object is the product of the mass of the object and its velocity.
A force is equal to the rate of change of momentum of the object upon which it acts.
The total momentum of a system of interacting objects remains constant, providing no resultant external force acts on the system.
In a perfectly elastic collision, the relative momentum of the objects before impact is equal to their relative momentum after impact.

10.

Which statement about collisions is correct?

Momentum is only conserved in perfectly elastic collisions.
When two objects of different masses collide, they exert forces of different magnitudes on each other.
Kinetic energy is conserved in all collisions.
The relative speed of approach is equal to the relative speed of separation for perfectly elastic collisions.

11.

Two bar magnets P and Q are mounted on floats which can slide without friction along an air track. The two magnets slide towards each other along the air track and interact, without making contact.

The relative speed of approach of the magnets is equal to their relative speed of separation. Which statement about P and Q must be correct?

During the interaction between P and Q some of the total momentum is lost.
During the interaction between P and Q some of the total kinetic energy is lost.
The values of (kinetic energy of P + kinetic energy of Q) before and after the interaction are equal.
The momentum of Q after the interaction is equal to the momentum of P before the interaction.

12.

Which quantities are conserved in an inelastic collision?

	kinetic energy	total energy	linear momentum
A	conserved	not conserved	conserved
B	conserved	not conserved	not conserved
C	not conserved	conserved	conserved
D	not conserved	conserved	not conserved

13.

Two balls, X and Y, approach each other along the same straight line and collide. The collision is perfectly elastic. Their initial speeds are and respectively. After the collision they move apart with speeds and respectively. Their directions are shown.

Which equation is correct?

14.

Skaters of masses and move directly towards each other and collide. Before the collision, the heavier skater is moving to the right at a speed of and the lighter skater is moving to the left at a speed of . After the collision, the heavier skater moves to the right at a speed of . What is the relative speed of separation of the two skaters?

15.

Two objects X and Y in an isolated system undergo a perfectly elastic collision. The velocities of the objects before and after the collision are shown.

What is the speed v of Y after the collision?

16.

The diagram shows a particle P, travelling at speed v, about to collide with a stationary particle Q of the same mass. The collision is perfectly elastic.

Which statement describes the motion of P and of Q immediately after the collision?

P comes to rest and Q acquires speed .
P rebounds with speed and Q remains stationary.
P and Q both travel in the same direction with speed .
P rebounds with speed and Q acquires speed .

17.

A helium atom of mass collides normally with a wall. The atom arrives at the wall with speed and then rebounds along its original path. Assume that the collision is perfectly elastic. What is the change in the momentum of the atom during its collision?

zero

18.

Which row in the table gives the quantities that are conserved in a perfectly elastic collision between two gas molecules?

	total momentum	total kinetic energy
A	conserved	conserved
B	conserved	not conserved
C	not conserved	conserved
D	not conserved	not conserved

19.

Two balls X and Y are moving towards each other with speeds of and respectively.

They make a perfectly elastic head-on collision and ball Y moves to the right with a speed of . What is the speed and direction of ball X after the collision?

to the left
to the left
to the right
to the right

20.

Two spheres approach each other along the same straight line. Their speeds are and before they collide. After the collision, the spheres separate with speeds and in the directions shown below.

The collision is perfectly elastic. Which equation must be correct?

21.

Two bodies travelling along the same straight line collide in a perfectly elastic collision. Which statement must be correct?

The total momentum is conserved but the total kinetic energy will be reduced.
The initial speed of one body will be the same as the final speed of the other body.
One of the bodies will be stationary at one instant.
The relative speed of approach between the two bodies equals their relative speed of
separation.

22.

Which statement about a perfectly elastic collision between two bodies in an isolated system is correct?

Both total kinetic energy and total momentum are conserved.
Total momentum is conserved, but total kinetic energy is not conserved.
Total kinetic energy is conserved, but total momentum is not conserved.
Neither total kinetic energy nor total momentum is conserved.

23.

What is a statement of the principle of conservation of momentum?

Momentum is the product of mass and velocity.
In an elastic collision momentum is constant.
The momentum of an isolated system is constant.
The force acting on a body is proportional to its rate of change of momentum.

24.

Two frictionless trolleys are moving towards each other along the same horizontal straight line. Their masses and velocities are shown.

The trolleys collide and stick together. What is the velocity of the trolleys after the collision?

to the right
to the right
to the left
to the left

25.

Two solid spheres form an isolated system. Sphere X moves with speed in a straight line directly towards a stationary sphere Y, as shown.

The spheres have a perfectly elastic collision. After the collision, sphere X moves with speed in the same direction as before the collision. What is the speed of sphere Y?

26.

Two railway trucks of masses and move towards each other in opposite directions with speeds and respectively. These trucks collide and stick together. What is the speed of the trucks after the collision?

27.

A wooden hollow cylinder with a radius of 1 m can rotate around a horizontal geometric axis. The moment of inertia of the cylinder is . The cylinder is stationary at first. A bullet with a mass of 10 g approaches at a speed of 850 m / s and sinks at an angle into the circumference of the cylinder. At what angular velocity does the cylinder rotate?

28.

Two cylinders with the same radii of 0.2 m rest on the same axis of rotation and are initially spaced in the vertical plane. The lower cylinder weighs 2 kg and the upper 1 kg. Initially, only the lower cylinder rotates at an angular velocity of 12 rad/s. At some point, the upper cylinder falls on the lower one. The cylinders initially slip until they both begin to rotate at the same angular velocity.

What is the angular velocity of both cylinders after the coupling?
What is the angular impulse with which the lower cylinder acts on the upper during coupling?

29.

On a large round plate weighing 300 kg stands a man weighing 60 kg. The plate can rotate around the vertical axis without friction. In the beginning, the plate and the man are at rest. Suddenly, the man begins to move around the perimeter of the plate counterclockwise and moves through the whole perimeter of the plate in 6 s. At what angular velocity does the plate rotate?

30.

A small stone attached to a thin string circulates in a circular path with a radius of 80 cm. Its linear speed is . During rotation, the cord hits a rod of radius 5 mm at a distance of 50 cm from the axis of rotation. What is the linear velocity of the stone when the string is wrapped around the rod three times?