2.5 - Momentum

Momentum, Mass and Velocity

Momentum, $p$ , is the product of mass and velocity; it is a vector in the direction of velocity.

p = m v

S.I. unit: $kg m s^{- 1}$ , equivalent to $N s$ .
A larger mass or greater velocity produces a larger momentum. Objects with larger momentum are harder to stop.
A marble released from higher on a slope moves faster and displaces a cardboard target farther. A heavier marble also displaces it farther; the displacement represents the difficulty in stopping the marble.
Lorry: $m = 20000 kg$ , $v = 22 m s^{- 1}$ ; $p = 440000 kg m s^{- 1}$ .
Car: $m = 2000 kg$ , $v = 30 m s^{- 1}$ ; $p = 60000 kg m s^{- 1}$ .
At the car’s velocity, the lorry has $p = (20000) (30) = 600000 kg m s^{- 1}$ .

Principle of Conservation of Momentum: total momentum before a collision equals total momentum after it when no external force acts on the system.

m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}

$m_{1}, m_{2}$ are the two masses; $u_{1}, u_{2}$ are their initial velocities; $v_{1}, v_{2}$ are their final velocities. Include the direction of each velocity through its sign.
When two trolleys join after collision, they have a common velocity, $v$ , and combined momentum $(m_{1} + m_{2}) v$ .
Small differences between measured momentum before and after a trolley collision can result from external friction that is not fully compensated.
Fire hose: water leaving at high speed has large forward momentum. Firefighters experience a backward recoil and must hold the hose firmly; two or more may be needed.

An explosion separates an initially stationary object into two or more parts.
For a closed system without external forces, total momentum is conserved. If initial momentum is zero:

0 = m_{1} v_{1} + m_{2} v_{2} \Rightarrow m_{1} v_{1} = - m_{2} v_{2}

The parts have equal and opposite momenta.
Rocket launch illustrates this principle: expelled hot gases move downwards while the rocket moves upwards.