Linear Momentum and Impulse

A micro-course in Linear Momentum and Impulse that is the section four of the course in classical mechanics available on Physics 12: Classical Mechanics. This course contains lectures about linear momentum and impulse, conservation law of linear momentum, and elastic and in-elastic collisions. Problems and solutions are also included to this course.

Linear Momentum

Linear momentum of an object is the product of its mass and velocity:   

The instantaneous change of linear momentum of an object at a given instant of time, is equal to the net force acting on the object at that time, i.e.,

 This equation is another formalism of Newton's second law. 

Impulse

Impulse is defined as the product of force and time interval in which the force acts on an object. If the force varies during the time interval, its average value over time interval, times the time interval, is the impulse acting on the object during the time interval.

 If the force acting on the object is the net force acting on it, impulse is equal to the difference of the final and initial momentum of the object.

Conservation Law of  Linear Momentum

Linear momentum of a system is conserved, if the net external force acting on it, is equal to the zero. 
According to the conservation law of linear momentum, the sum of linear momentum of objects involved in a collision, are the same, before and after the collision, if the system of objects involved in the collision is isolated (i.e., there isn't any external force acting on the system). The conservation law of linear momentum of an isolated system comes from newton's second and third laws. This is discussed
in Grade-12 Physics: Mechanics.

Elastic and Inelastic Collisions

If the system of objects involved in a collision is isolated (or there isn't any external force acting on it), the collision in it the kinetic energy is conserved, is said to be elastic; when the kinetic energy is not conserved, collision is inelastic.
So, with elastic collisions, both the momentum and kinetic energy are conserved. For instance, if two particles collides elastically, the conservation laws of linear momentum and energy can be written as,

where indices 1 and 2 refer to particles 1 and 2, and 'i' and 'f' refers to initial and final values.
For inelastic collisions only momentum is conserved (if the system is isolated). So, only the second equation is valid for the inelastic collision of two particles.

Problems and Solutions

Problem 1

A car with mass of 800 kg, moving at 10 m/s crashes into a barrier and stops in 0.05 s.
a. What is the impulse needed to stop the car?
b. What is the average force acted on the car? 

Problem solving lecture in momentum and impulse:1

Problem 2
 
A billiard ball with initial speed of 5 m/s collides with another billiard ball with identical mass that’s initially at rest. After the collision, the first ball bounces off  at 30° to the left of its original direction, with speed of  2 m/s; the second ball moves off at angle θ  to the  right of  the first ball’s original direction (see figure).
Neglect friction and find the direction and speed of second ball after collision.
Find kinetic energy of these two balls before and after collision, if the mass of each ball is 0.8 kg. Is the collision elastic? 

Problem solving lecture in momentum and impulse:2