Collisions (3.5.2)
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The principle of conservation of momentum states that the total momentum of a closed system remains constant. A closed system is one in which no external forces act upon it.
A system can consist of many objects that interact with each other:
- The objects can interact through contact forces, like snooker balls bouncing off each other.
- Objects can also interact through non-contact forces, such as the electrostatic repulsion between two electrons or the gravitational attraction between planets.
In order for the principle of conservation of momentum to be obeyed, the internal forces of a closed system must not change the total momentum. This is a consequence of Newton’s laws of motion.
Newton’s third law states that if object A exerts a force on object B, object B will exert an equal and opposite force on object A. For example, if you push against a wall, the wall will push back against you with a force of equal magnitude, so that it remains stationary:
Newton’s second law states that the net force is equal to the change in momentum over a time period
Therefore, if the net force equals zero, then the change in momentum equals zero, and momentum is conserved.
The principle of conservation of momentum in a closed system can be stated mathematically as:
Momentum is a vector quantity. Therefore, both the magnitude and direction of the momentum vector are conserved for a closed system.
In one dimension, the direction of the momentum vector does not change after an interaction, since the objects move along one direction. For example, consider the head-on collision of Ball A and Ball B of equal mass :
The momentum of an object with mass moving at speed is equal to:
Therefore, the principle of conservation of momentum for this collision leads to the expression:
Where is the final speed of Ball A. Cancelling the masses and rearranging gives:
If a collision between two objects occurs at an angle in two-dimensions, their final velocities will be in different directions.
Ball A moves at to the right, and collides at an angle with a stationary Ball B of equal mass; they move off in directions to each other.
Since momentum is conserved, the momentum vectors of the balls after the collision must sum to the momentum vector of Ball A before the collision.
The masses are the same, so the velocity vector triangle is the same shape as the momentum vector triangle. The vector triangle is a right-angle triangle, so the final velocity of Ball B can be found from Pythagoras’ theorem:
Therefore:
Conservation of momentum requires that momentum is conserved in any direction.
The diagram illustrates a collision between Ball A of mass and initial velocity along the direction, colliding with Ball B of mass The collision results in final velocities and at angles of and to the horizontal.
Conservation of momentum in the direction leads to:
Since there is no initial momentum in the direction, the expression for the direction is:
In a collision, both the total momentum and the total energy are conserved. However, the kinetic energy before the collision can be converted to other forms of energy, such as heat or sound.
- Perfectly elastic collisions: both the total momentum and the total kinetic energy are conserved.
- None of the kinetic energy is converted to other forms.
- Inelastic collisions: the total momentum is conserved, but the total kinetic energy is not.
- Some of the kinetic energy is converted to other forms.
Question walkthrough
Proving a Collision is Elastic
Elastic collision
Question walkthrough
Classifying a Collision as Elastic or Inelastic
Determining collision type