There are two kinds of momentum, linear and angular. A spinning object has angular momentum; an object traveling with a velocity has linear momentum. We'll focus on linear momentum for now, and just refer to it as momentum.

The symbol for momentum is p.

Four fast facts about momentum

1. p = mv
   
   
2. Momentum is a vector, pointing in the direction of the velocity.
   
   
3. If there is no net force acting on a system, the system's momentum is conserved.
   
   
4. A net force produces a change in momentum that is equal to the force multiplied by the time interval during which the force was applied.

   Happy and Sad Balls

   Two balls are rolled down an incline, one at a time, and strike a wooden block that is standing upright. The Happy ball is a regular bouncy rubber ball, which rebounds from the block with a speed close to what it had before impact. The second Sad ball stops dead on impact. One ball knocks the block over - which is it?

   1. The Happy (bouncy) ball
   2. The Sad (non-bouncy) ball
   
   

   The Sad ball experiences a change in momentum of -mv, going from mv to zero, so it must impart a momentum of mv to the block. The momentum of the Happy ball goes from +mv to -mv, a change of -2mv. It gives 2mv of momentum to the block, twice as much as the Sad ball.

   Impulse

   Newton originally wrote his Second Law of Motion in terms of momentum, as what we now call the impulse equation. Impulse is the product of a force and the time interval over which the force acts.

   Remember that work (a force acting over a distance) produces a change in kinetic energy. An impulse (a force acting over a time interval) produces a change in momentum.

   F = ma

   F = m dv/dt = dp/dt

   Expressed as an integral, this becomes:

   ∫ F dt = Δp

   The impulse, which is the change in momentum, is the area under the force vs. time graph.

   Impulse examples

   A hose sprays water directly at a wall. If the volume of water emerging from the hose is X liters/second, and the water has velocity v, directed horizontally, how much force is exerted on the wall by the water?

   Assume that the water does not bounce back, but is simply stopped by the wall. Focus on one second worth of water. X liters has a mass of X kg, so the momentum of the water goes from Xv to zero, a change of -Xv. To produce this change in momentum, the wall must exert a force on the water of -Xv newtons. The water exerts an equal and opposite force on the wall, Xv newtons in the direction the hose points.

   If v = 10 m/s and X is 3 liters/second, the force has a magnitude of 30 N.

   A mass moving in the x direction at constant speed on a frictionless table is subjected to a force in the y-direction. The force is applied for 0.5 seconds starting at t=2 seconds. Which of the following situations corresponds to the motion of the mass?