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Variable Mass

Variable Mass. After the split, the sum of momentum is conserved. P = m 1 v 1 + m 2 v 2 Center of mass velocity remains the same. The kinetic energy is not conserved. Before the split, momentum is P = MV M total mass V center of mass velocity. Break Up. v 1. V. V. M. v 2.

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Variable Mass

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  1. Variable Mass

  2. After the split, the sum of momentum is conserved. P = m1v1 + m2v2 Center of mass velocity remains the same. The kinetic energy is not conserved. Before the split, momentum is P = MV M total mass V center of mass velocity Break Up v1 V V M v2

  3. A 325 kg booster rocket and 732 kg satellite coast at 5.22 km/s. Explosive bolts cause a separation in the direction of motion. Satellite moves at 6.69 km/s Booster moves at 1.91 km/s Find the kinetic energy before separation, and the energy of the explosion. Kinetic energy before separation is (1/2)MV2 K = (1/2)(1057 kg)(5.22 x 103 m/s)2 = 14.4 GJ Kinetic energy after separation K1 = 16.4 GJ K2 = 0.592 GJ The difference is the energy of the explosion. Kint = 2.6 GJ Explosions

  4. Change in Momentum • The law of action was redefined to use momentum. • The change can be due to change in velocity or a change in mass

  5. Infinitessimal Change • In a short time the following happens: • The mass goes from m to m + Dm • The velocity goes from v to v + Dv • The mass added or removed had a velocity u compared to the object • Momentum is conserved

  6. Thrust • If there is no external force the force to be applied must be proportional to the time rate of change in mass. The force -u(dm/dt) is the thrust

  7. 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. Water Force

  8. Rocket Speed • Rockets decrease their mass, so we usually write the mass change as a positive quantity. • The equation can be integrated to get the relationship between the mass and increased velocity.

  9. Applied Force • If there is an external force that must equal the time rate of change in momentum. • Force is needed to maintain the speed.

  10. 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. Heavy Water

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