Momentum Conservation

Law of Action Redefined Newton originally framed the second law (acceleration) in terms of momentum, not velocity. The rate of change in momentum is proportional to the net force and the change is in the same direction as the force Final momentum: changed by the force Rocket: has momentum Initial momentum Force: changes momentum

Momentum Change If two forces are equal in magnitude, then the changes in momentum are equal in magnitude. F12 pf1 pi2 pi1 pf2 F21

Internal Forces When two bodies strike each other they exert a force on each other. Considered together they are called a system. The internal forces cancel out from the law of reaction. Equal and opposite The external forces remain. F21 F12 Fg1 Fg2

Isolated System A system with no external forces is isolated. Only internal forces that cancel out The net change in momentum for all internal forces must be zero. Total momentum in an isolated system is constant. Zero change means constant value

Separation A 732 kg satellite attached to a 325 kg booster rocket coast at 5.22 km/s. Explosive bolts cause a separation in the direction of motion propelling the satellite at 6.69 km/s. Find the momentum of the booster and the amount of energy in the explosion. V M v1 v2 m1 m1

Break Up Before the split, momentum is P = MV M total mass V initial velocity After the split, the sum of momentum is conserved. P = m1v1 + m2v2 Momentum is conserved. Initial conditions: M=732 kg+325 kg=1057 kg V = 5.22 km/s = 5220 m/s P = 5.52 x 106 kg m/s Final satellite momentum: m1 = 732 kg v1 = 6.69 km/s p1 = 4.90 x 106 kg m/s Final booster momentum p2 = 0.62 x 106 kg m/s m2 = 325 kg v2 = 1.91 km/s

Explosion Before the split, kinetic energy is Ki = (1/2)MV2 M total mass V inital velocity After the split, the explosion adds kinetic energy. Kf = (1/2)m1v12 + (1/2)m2v22 The kinetic energy is not conserved, but energy is. Initial conditions: M = 1057 kg V = 5.22 km/s = 5220 m/s Ki = 14.4 GJ Final energies: K1 = 16.4 GJ K2 = 0.593 GJ Kf = 17.0 GJ Explosion energy: KX = 2.6 GJ

Law of Reaction Redefined The law of reaction can be defined in terms of momentum. In an isolated system the total momentum is conserved. total momentum of the rocket and gas stays the same rocket hot gas is forced out rocket momentum increases reaction force acts on the rocket

Thrust If there is no external force the force to be applied must be proportional to the time rate of change in mass. The mass changes by Dm The velocity changes by Dv The mass added or removed had a velocity u compared to the object The force u(Dm/Dt) is the thrust

Water Force Thrust can be used to find the force of a stream of water. A hose provides a flow of 4.4 kg/s at a speed of 20. m/s. The momentum loss is (20. m/s)(4.4 kg/s) = 88 N The momentum loss is the force.

Heavy Water Water is poured into a beaker from a height of 2 m at a rate of 4 g/s, into a beaker with a 100. g mass. What does the scale read when the water is at 200. ml in the beaker (1 ml is 1 g)? Answer: 302 g There is extra momentum from the falling water. This is about 0.024 N or an equivalent mass of 2.4 g.