Conservation of Momentum
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1 Conservation of Momentum
2
3 Law of Conservation of Momentum The sum of the momenta before a collision equal the sum of the momenta after the collision in an isolated system (=no external forces acting).
4 Law of Conservation of Momentum: The total momentum of an isolated system of bodies remains constant. (Isolated system: meaning that all forces acting on the bodies are included and the sum of the external forces applied to the system is zero. External forces like F f or F g.) Momentum before = Momentum after m 1 v 1 + m 2 v 2 = m 1 v' 1 + m 2 v' 2 (Elastic Collision) m 1 v 1 + m 2 v 2 = (m 1 + m 2 )v (Inelastic Collision) v = velocity before collision v' = velocity after collision
5 Elastic Collisions!
6 Elastic Collision Elastic Collisions: Two or more objects collide, bounce (don t stick together), and kinetic energy is conserved. An ideal situation that is often never quite reached billiard ball collisions are often used as an example of elastic collisions. Kinetic (motion) energy is conserved: KE 1 + KE 2 = KE' 1 + KE' 2 ½m 1 v ½m 2 v 2 2 = ½m 1 v' ½m 2 v' 2 2 Momentum is conserved: m 1 v 1 + m 2 v 2 = m 1 v' 1 + m 2 v' 2
7 Inelastic Collision Inelastic Collision: two or more objects collide and do not bounce off each other, but stick together. Or, an explosion where one object starts w/one momentum and then separates into two or more objects w/separate final momenta. Kinetic energy is not conserved. KE before = KE after + heat + sound + etc. The kinetic energy lost is transformed into other types of energy, but Total energy is always conserved! m 1 v 1 + m 2 v 2 = (m 1 + m 2 )v'
8 Elastic vs. Inelastic Collisions (a) A hard steel ball would rebound to its original height after striking a hard marble surface if the collision were elastic. (b) A partially deflated basketball has little bounce on a soft asphalt surface. (c) A deflated basketball has no bounce at all. Show happy balls
9 A ball of mass kg and velocity m/s collides head on with a second ball of mass kg that is initially at rest. No external forces act on the balls. If the balls collide and bounce off one another, and the second ball moves with a velocity of m/s, determine the velocity of the first ball after the collision, including direction.
10 Did you get? v f1 = m/s (ball 1 rebounds)
11 Lets prove if energy is conserved.. Remember from a previous slide that for Elastic Collisions: Two or more objects collide, bounce (don t stick together), and kinetic energy is conserved. KE 1 + KE 2 = KE' 1 + KE' 2 ½m 1 v 12 +½m 2 v 22 =½m 1 v' 12 +½m 2 v' 2 2 Apply this equation to the last problem and see if it is true.
12 Energy is conserved!
13 Question: The total momentum of a system is conserved A. always. B. if no external forces act on the system. C. if no internal forces act on the system. D. never; momentum is only approximately conserved.
14 Answer: The total momentum of a system is conserved A. always. B. if no external forces act on the system. C. if no internal forces act on the system. D. never; momentum is only approximately conserved.
15 Question: In an inelastic collision, A. impulse is conserved. B. momentum is conserved. C. force is conserved. D. Kinetic energy is conserved. E. elasticity is conserved.
16 Answer: In an inelastic collision, A. impulse is conserved. B. momentum is conserved. C. force is conserved. D. Kinetic energy is conserved. E. elasticity is conserved.
17
18 Forces During a Collision Slide 9-20
19 The Law of Conservation of Momentum In terms of the initial and final total momenta: In terms of components:
20
21 Example Problem A curling stone, with a mass of 20.0 kg, slides across the ice at 1.50 m/s. It collides head on with a stationary kg hockey puck. After the collision, the puck s speed is 2.50 m/s. What is the stone s final velocity? Slide 9-23
22 Answer: 1.48 m/s
23 Rocket propulsion is an example of conservation of momentum: The rocket doesn t push on the environment. The rocket pushes the exhaust gas in one direction (backward), and the exhaust gas pushes the rocket in the opposite direction (forward). Newton s third law, the force and time acting on the rocket and the gas (as a whole) are equal and opposite. The momentum is conserved. The momentum before is zero and the momentum after is a total of zero. Positive momentum of the rocket = Negative momentum of the gas. Slide 9-24
24 Inelastic Collisions: For now, we ll consider perfectly inelastic collisions: A perfectly inelastic collision results whenever the two objects move off at a common final velocity.
25 Example Problem Jack stands at rest on a skateboard. The mass of Jack and the skateboard together is 75 kg. Ryan throws a 3.0 kg ball horizontally to the right at 4.0 m/s to Jack, who catches it. What is the final speed of Jack and the skateboard?
26 Answer: m/s
27 Recall from a previous slide Inelastic Collision: two or more objects collide and do not bounce off each other, but stick together. Or, an explosion where one object starts w/one momentum and then separates into two or more objects w/separate final momenta. Kinetic energy is not conserved. KE before = KE after + heat + sound + etc. The kinetic energy lost is transformed into other types of energy, but Total energy is always conserved! m 1 v 1 + m 2 v 2 = (m 1 + m 2 )v'
28 Prove if the last problem conserved energy. KE 1 + KE 2 = KE 1+2 ½ m 1 v ½ m 2 v 2 2 = ½ m 1+2 v' 2 Apply this equation to the last problem and see if it is true.
29 Energy is not conserved!
30 A 20.0 g ball of clay traveling east at 2.00 m/s collides with a 30.0 g ball of clay traveling 30.0 o south of west at 1.00 m/s. The two pieces stick together and become one. What are the speed and direction of the final piece of clay? Momentum is a vector including direction. Hint: Draw your vectors tip to tail and draw the resultant momentum vector (p final). Resolve all vectors into x and y-components. Determine the sum of the x-components and the sum of the y-components and draw your final resultant vectors making a right triangle. Solve for p final and angle. Solve for v final.
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