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Understanding Variable Mass and Conservation of Momentum in Physics

This text explains the principles of variable mass in physics, focusing on conservation of momentum, kinetic energy, explosive energy calculations, and practical applications like rocket propulsion and forces involved in streams of water. It also touches on infinitesimal changes, thrust, rocket speed, applied force, and a practical scenario involving water mass calculations.

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Understanding Variable Mass and Conservation of Momentum in Physics

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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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