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MOMENTUM AND IMPULSE

 

Linear Momentum:  It is the product of the mass of a moving object and its velocity.  Velocity is constant in linear momentum.  Momentum is a vector quantity with the direction same as the velocity.


 

momentum (kg.m/s)
or (N.s)

vector quantity with the direction the same as the velocity
 

m:

mass

(kg)


scalar quantity
 
velocity (m/s)
vector quantity
 


Impulse:

Impulse is the change in momentum.  Impulse is also the integral of force over time, it is measured in Newton-seconds.   Unit of impulse the same as momentum
(kg.m/s) or (N.s).


 

∆t:  Time interval over which the force acts (s)

∑F:  Average force acting on the object over the time interval ∆t (N)

 

Momentum and impulse in two dimensions:

x component of the momentum:
 

 
y component of the momentum:
 

 

x component of the impulse:
 

Δpx = ΣFx. Δt
 
y component of the impulse:
 
Δpy = ΣFy. Δt
 

 

Conservation of Momentum:

If the net force acting on a system of interacting objects is zero, then the linear momentum of the system before the interaction equals the linear momentum of the system after the interaction.

Momentum is conserved both in elastic and inelastic collision.

Conservation of Momentum for colliding objects:

m1.v1 + m2.v2... = m1.v'1 + m2.v'2...

m1:  Mass of the fist object
m2:  Mass of the second object
v1:  Velocity of the fist object before collision
v2:  Velocity of the second object before collision
v'1:  Velocity of the first object after collision
v'2:  Velocity of the second object after collision

Conservation of Momentum in two dimensions for interacting objects have also same rule, and this rule applies for x component of the momentum and y component of the momentum.

m1.v1x + m2.v2x...= m1.v'1x + m2.v'2x...     (x component of the momentum)

m1.v1y + m2.v2y... = m1.v'1y + m2.v'2y...     (y component of the momentum)

 

Conservation of Kinetic Energy:

Kinetic energy is conserved in elastic collisions only.  Kinetic energy is not conserved in inelastic collisions.

For elastic collision:

0.5 m1.v12 + 0.5 m2.v22 = 0.5 m1.v'12 + 0.5 m2.v'22


Inelastic Collision

Momentum is conserved in all kinds of collisions such as elastic, inelastic and completely inelastic collisions in an isolated system.

Kinetic energy is not conserved in inelastic collisions.  Total kinetic energy after the collision is different from the total kinetic energy before collision.  In completely inelastic collision, the objects stick together and the decrease in total kinetic energy will be maximum.

Most of the inelastic collisions, the total kinetic energy after the collision is less than the the total initial kinetic energy before the collisions.  However, some inelastic collisions, such as an explosion, the total final kinetic energy of the system is greater than the initial kinetic energy of the system because kinetic energy produced during the explosion.

A good example of an inelastic collision is the collisions of tennis balls.

If the two bodies stick together after the collision, the collision is said to be completely inelastic collision.  A bullet embedding itself in a block of wood is an example of inelastic collision.  Another example for completely inelastic collision would be two cars that crash and lock bump.  A meteorite collides head-on with the the Earth is also completely inelastic collision.
 

Elastic Collision

Momentum is conserved in all kinds of collisions such as elastic, inelastic and completely inelastic collisions in an isolated system.

Kinetic energy is conserved only in elastic collisions.  Total kinetic energy after the collision equals total kinetic energy before the collision.

In real life, it is very difficult to produce completely elastic collisions.  The only really good examples of elastic collisions are collisions of atomic particles.  Collisions between hard steel balls as in the swinging balls apparatus are nearly elastic.  The collisions of pool balls, or a superballs are also nearly elastic.  The collisions of air molecules with the walls of container are elastic collision.

Calculations for One-dimensional Elastic Collision:

Equation 1:   m1.v1 + m2.v2 = m1.v'1 + m2.v'  (Momentum is conserved in all kinds of collisions)

Equation 2:   0.5 m1.v12 + 0.5 m2.v22 = 0.5 m1.v'12 + 0.5 m2.v'22   (Kinetic energy is conserved only in elastic collisions)

If the collision is elastic, and if the velocity of the second object before collision equals to zero, we can get the following equations by using above equation 1 and equation 2:

m1(v1- v'1) =  m2 - v'2

m1(v12 - v'12) = m2.v'22

v1 + v'1 = v'

v'1 = v1 (m1 - m2) / (m1 + m2)

v'2 = v1 (2m1) / (m1 + m2)

If the collision is elastic, but the velocity of the second object before collision does not equal to zero, we should make adjustment to make the velocity of the second object before collision equals to zero.  For example:

v1 = 5 m/s
v2 =  3 m/s

Subtract 3 m/s from both velocities.  In this case:

v1 = 2 m/s
v2 =  0

After you complete all the calculations, you add 3 m/s to all the velocities.