PRINCIPLE OF CONSERVATION OF MOMENTUM

All bodies in the nature interact with each other. However in many cases the external forces exerted on the considered system of interacting bodies by other bodies are too small or balance each other, so they can be ignored. Such situations allow considering interaction of two or several bodies only with each other. For this purpose the concept of an isolated system of bodies is introduced in physics.

System of bodies where the bodies interact only among themselves without external forces exerted on them by outside bodies is called an isolated system.

For example, when two balls collide on a table, the force of gravity doesn't influence their motion, because it is balanced by the force of the table reaction and so on.

Let the balls to have masses m₁ and m₂, and their velocities to be and respectively (fig.67).

The forces of interaction, whatever their nature, obey Newton’s third law:

,

where Dt is the time interval of interaction, are the velocities of bodies after their interaction.

Fig.67

Let's rearrange the above equation to have on its left side the vector sum of momenta of bodies before interaction and on its right side the vector sum of momenta of bodies after interaction:

.

During the interaction of bodies their individual momenta can change but the total momentum of the system of bodies is constant. Hence,

the total momentum of an isolated system of bodies interacting with each other is constant:

.

This is one of the basic laws of nature known as the principle of conservation of momentum.

We can generalize this principle for a system that contains any number of bodies interacting only with each other, i.e. for an isolated system of bodies. But real systems are not isolated. However, in many cases when external forces are negligibly small compared with the internal forces between bodies the principle of conservation of momentum can be used.

Self-testing questions

1. What systems are called isolated?

2. What is the statement of the principle of conservation of momentum?

3. How does the principle of conservation of momentum work during the collision of bodies? Give examples.

4. Under what conditions is the principle of conservation of momentum for the system of interacting bodies valid?

Exercise 17

1. What is the momentum of a person with mass 60 kg that runs at a velocity of 6 km/h?

2. A body with mass 2 kg moving along a straight line at a velocity of 20 m/s collides with another body which has mass 3 kg and is initially at rest. Then they start moving together at velocity u. Neglecting the effects of external forces, determine:

a) the total momentum of bodies before collision;

b) the total momentum and velocity after collision.

3. The boy with mass 50 kg runs at a velocity of 5 m/s. He jumps on a cart with mass 100 kg moving at a velocity of 2 m/s. What is the velocity of the cart right after the jump of the boy on the cart?

4. A shell is fired vertically upward. At the highest point of its trajectory the shell explodes into two equal fragments, one of which moves upward and another moves downward. If the first fragment falls on the second one with a speed of u then with what speed the second fragment falls on the ground?

Date: 2015-01-12; view: 989