Conservation of momentum

Consider two interacting objects.  If object 1 pushes on object 2 with a force F = 10 N for 2 s to the right, then the momentum of object 2 changes by 20 Ns = 20 kgm/s to the right.  By Newton's third law object 2 pushes on object 1 with a force F = 10 N for 2 s to the left.  The momentum of object 1 changes by 20 Ns = 20 kgm/s to the left.  The total momentum of both objects does not change.  For this reason we say that the total momentum of the interacting objects is conserved. 

Newton's third law implies that the total momentum of a system of interacting objects not acted on by outside forces is conserved.

The total momentum in the universe is conserved.  The momentum of a single object, however, changes when a net force acts on the object for a finite time interval.  Conversely, if no net force acts on an object, its momentum is constant.

For a system of objects, a component of the momentum along a chosen direction is constant, if no net outside force with a component in this chosen direction acts on the system.

Problem:

A 0.1 kg ball is thrown straight up into the air with an initial speed of 15 m/s.  Find the momentum of the ball
(a)  at its maximum height and
(b)  half way up to its maximum height.

Solution:

imageProblem:

Two blocks of mass M and 3 M are placed on a horizontal frictionless surface.  A light spring is attached to one of them, and the blocks are pushed together with the spring between them.  A cord holding them together is burned, after which the block of mass 3 M moves to the right with a speed of 2 m/s.
(a)  What is the speed of the block of mass M?
(b)  Find the original elastic potential energy in the spring if M = 0.35 kg.

Solution:


Collisions

imageIn collisions between two isolated objects Newton's third law implies that momentum is always conserved.  In collisions, it is assumed that the colliding objects interact for such a short time, that the impulse due to external forces is negligible.  Thus the total momentum of the system just before the collision is the same as the total momentum just after the collision. 
Collisions in which the kinetic energy is also conserved, i.e. in which the kinetic energy just after the collision equals the kinetic energy just before the collision, are called elastic collision.  In these collisions no ordered energy is converted into thermal energy. 
Collisions in which the kinetic energy is not conserved, i.e. in which some ordered energy is converted into internal energy, are called inelastic collisions.  If the two objects stick together after the collision and move with a common velocity vf, then the collision is said to be perfectly inelastic.

Note:
In collisions between two isolated objects momentum is always conserved.
Kinetic energy is only conserved in elastic collisions.

We always have m1v1i + m2v2i = m1v1f + m2v2f.
Only for elastic collisions do we also have ½m1v1i2 + ½m2v2i2 = ½m1v1f2 + ½m2v2f2.

Problem:

If two objects collide and one is initially at rest, is it possible for both to be at rest after the collision?  Is it possible for one to be at rest after the collision?  Explain!

Solution:

Problem:

A 10 g bullet is stopped in a block of wood (m = 5 kg).  The speed of the bullet-wood combination immediately after the collision is 0.6 m/s.  What was the original speed of the bullet?

Solution:

Problem:

Two cars of equal mass and equal speeds collide head on.  Do they experience a greater force if the collision is elastic or perfectly inelastic and they stick together?

Solution:

Problem:

A 90 kg fullback running east with a speed of 5 m/s is tackled by a 95 kg opponent running north with a speed of 3 m/s.  If the collision is perfectly inelastic, calculate the speed and the direction of the players just after the tackle.

Solution:

Problem:

A 30,000 kg freight car is coasting at 0.850 m/s with negligible friction under a hopper that dumps 110,000 kg of scrap metal into it.  What is the final velocity of the loaded freight car?

Solution:

Problem:

A neutron in a reactor makes an elastic head-on collision with the nucleus of a carbon atom initially at rest.
(a) What fraction of the neutron's kinetic energy is transferred to the carbon nucleus?
(b) If the initial kinetic energy of the neutron is 1.6*10-13 J, find its final kinetic energy and the kinetic energy of the carbon nucleus after the collision.
(The mass of the carbon nucleus is about 12 times the mass of the neutron.)

Solution:

Problem:

After a completely inelastic collision between two objects of equal mass, each having initial speed v, the two move off together with speed v/3.  What was the angle between their initial directions?

Solution:

Problem:

The mass of the blue puck is 20% greater than the mass of the green one.  Before colliding, the pucks approach each other with equal and opposite momenta, and the green puck has an initial speed of 10 m/s.  Find the speed of the pucks after the collision, if half the kinetic energy is lost during the collision.

image

Solution:


Module 7: Question 1

During a visit to the International Space Station, an astronaut was positioned motionless in the center of the station, out of reach of any solid object on which he could exert a force.  Suggest a method by which he could move himself away from this position, and explain the physics involved.

Discuss this with your fellow students in the discussion forum!

Module 7: Question 2

Football coaches advise players to block, hit, and tackle with their feet on the ground rather than by leaping through the air.  Using the concepts of momentum, work, and energy, explain how a football player can be more effective with his feet on the ground.

Discuss this with your fellow students in the discussion forum!


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