# Conservation of Momentum

## Momentum of a body is defined as the product of its mass and velocity

The momentum of a body is defined as the product of its mass and velocity. Since velocity is a vector quantity - momentum is a vector quantity.

The momentum of a body can be expressed as

M = m v                                      (1)

where

M = momentum (kg m/s, lb ft/s)

m = mass (kg, lb)

v = velocity (m/s, ft/s)

The momentum of a body remains the same as long as there is no external forces acting on it. The principle of conservation of momentum can be stated as

the total linear momentum of a system is a constant

The total moment of two or more bodies before collision in a given direction is equal to the total momentum of the bodies after collision in the same direction and can be expressed as

MT = m1 v1 + m2 v2 + ..  + mn vn

=  m1 u1 + m2 u2 + ..  + mn un                                       (2)

where

v = velocity before collision (m/s, ft/s)

u = velocity after collision (m/s, ft/s)

### Example - Linear Momentum

A body with mass 30 kg and velocity 30 m/s collides with a body with mass 20 kg and velocity 20 m/s. The velocities of both bodies are in the same direction. Assuming the both bodies have same velocity after impact - the resulting velocity can be calculated as:

MT = (30 kg) (30 m/s)  + (20 kg) (20 m/s)

=  u ((30 kg) + (20 kg))

= constant

transformed:

u = (30 kg) (30 m/s)  + (20 kg) (20 m/s)  /  ((30 kg) + (20 kg))

= 26 (m/s)

### Example - the Recoil Velocity after a Rifle Shoot

A rifle weighing 6 lb fires a bullet weighing 0.035 lb at a muzzle speed of 2100 ft/s. The total moment of the rifle and bullet can be expressed as

MT = mr vr + mb vb

= mr ur + mb ub

The rifle and the bullet are initially at rest (vr = vb = 0):

MT = (6 lb) (0 ft/s) + (0.035 lb) (0 ft/s)

= (6 lb) ur (ft/s) + (0.035 lb) (2100 ft/s)

= 0

The rifle recoil speed:

ur = - (0.035 lb) (2100 ft/s) / (6 lb)

= - 12.25 (ft/s)

## Related Topics

• Mechanics - Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more

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