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Momentum, impulse, and collisions Chapter 8 Sections 1-5 Momentum The linear momentum of an object is defined as p mv Momentum has a magnitude of mv and the same direction as v Physics Chapter 8 2 Impulse When a particle is acted on by a constant force, the impulse of the force is defined as J Ft Physics Chapter 8 3 Impulse-momentum theorem v v0 at mv mv0 mat For a constant force mv mv0 Ft Ft mv mv0 Ft mv J p Physics Chapter 8 4 Example A 45-g golf ball initially at rest is given a speed of 25.0 m/s when a club strikes. If the club and ball are in contact for 2.00 ms, what average force acts on the ball? 562 N Physics Chapter 8 5 On your own A 0.160-kg hockey puck is moving on an icy, frictionless, horizontal surface. At t = 0 the puck is moving to the right at 3.00 m/s. Calculate the velocity of the puck after a force of 12.0 N directed to the left has been applied for 0.050 s. 0.75 m/s to the left Physics Chapter 8 6 Conservation of momentum If the vector sum of the external forces on a system is zero, the total momentum of the system is conserved. pbefore pafter p1 p2 Physics Chapter 8 7 Example A runaway 14,000 kg railroad car is rolling horizontally at 4 m/s toward a switchyard. As it passes by a grain elevator, 2000 kg of grain are suddenly dropped into the car. How long does it take the car to cover the 500-m distance from the elevator to the switchyard? Neglect friction and air drag. 143 s Physics Chapter 8 8 On your own During repair of the Hubble Space Telescope, an astronaut replaces a damaged solar panel. Pushing the detached panel away into space, she is propelled in the opposite direction. The astronaut’s mass is 60 kg and the panel’s mass is 80 kg. The astronaut is at rest relative to the spaceship when she shoves away the panel, and she shoves it at 0.3 m/s relative to the spaceship. What is her subsequent velocity relative to the spaceship? -0.4 m/s Physics Chapter 8 9 Components of momentum We can separate momentum into components, just like we can with force or velocity px mvx py mvy Physics Chapter 8 10 Example 30° A 90° B Physics Chapter 8 11 Example A 3.0-kg ball, A, is moving at a velocity of 5.0 m/s. It collides with a stationary ball, B, of mass 2.0 kg. After the collision, ball A moves off in a direction of 30° to the left of its original direction. Ball B moves off in a direction of 90° to the right of ball A’s final direction. Find the speeds of the balls after the collision. vA=4.33 m/s, vB= 3.75m/s Physics Chapter 8 12 On your own 30° A B Physics Chapter 8 45° 13 On your own A hockey puck B rests on a smooth ice surface and is struck by a second puck, A, which was originally traveling at 40.0 m/s and which is deflected 30.0° from its original direction. Puck B acquires a velocity at a 45.0° angle to the original direction of A. The pucks have the same mass. Compute the speed of each puck after the collision. vA=29.3 m/s vB=20.7 m/s Physics Chapter 8 14 Elastic collisions If all the forces acting during a collision are conservative, then no mechanical energy is lost or gained in the collision. When the total kinetic energy after the collision is the same as it was before the collision, then the collision was elastic. Collisions between billiard balls, marbles, or other similar objects are nearly elastic. Physics Chapter 8 15 Inelastic collisions Collisions between automobiles are one example of inelastic collisions. The kinetic energy is not conserved, because some energy goes into crumpling the cars. Collisions in which the two objects stick together afterwards are inelastic. Physics Chapter 8 16 Example In a feat of public marksmanship, you fire a bullet of mass mb into a hanging wood block of mass mw. The block, with the bullet embedded, swings upward. Noting the height, h, reached at the top of the swing, you inform the crowd of the bullet’s speed. How fast was the bullet traveling before it hit the block? ((m1+m2)/m1)*sqrt(2gh) Physics Chapter 8 17 On your own In Dallas, the morning after a winter ice storm, a 1400-kg automobile going west at 35.0 km/h collides with a 2800-kg truck going south at 50.0 km/h. If they become coupled on collision, what are the magnitude and direction of their velocity after colliding? 35.3 km/h @ 19.3 degrees west of south Physics Chapter 8 18 Elastic collisions For elastic collisions, both p and K are conserved. 1 2 mAvA1 mBvB1 mAvA2 mBvB 2 1 1 1 1 2 2 2 m Av A 2 mB vB 2 m Av A1 mB vB21 2 2 2 2 Physics Chapter 8 19 Elastic collisions 1 mBvB 2 mBvB1 mAvA1 mAvA2 1 mB vB 2 vB1 mA vA1 vA2 Physics Chapter 8 20 Elastic collisions 2 1 1 1 1 2 2 2 mB vB 2 mB vB1 m Av A1 m Av A2 2 2 2 2 2 2 mB vB2 2 vB21 mA vA21 vA2 2 2 mB vB 2 vB1 vB 2 vB1 mA vA1 vA2 vA1 vA2 Physics Chapter 8 21 Elastic collisions 2÷1 mB vB 2 vB1 vB 2 vB1 mA vA1 vA2 vA1 vA2 mB vB 2 vB1 mA vA1 vA2 vB 2 vB1 vA1 vA2 Physics Chapter 8 22 Be careful! The last equation is only true for elastic collisions. It is a special case, not the general case. For all collisions, momentum is conserved, but kinetic energy is only conserved for elastic collisions. Do not assume that a collision is elastic unless you are told that it is. Physics Chapter 8 23 On your own A neutron of mass mn and speed vn1 collides elastically with a carbon nucleus of mass mc initially at rest. What are the final velocities of both particles? Be careful! - The velocity equation developed for elastic collisions isn’t enough to solve the problem. It has 2 unknowns, so you need another equation. Use the conservation of momentum equation. Vn2=-((mc-mn)/(mn+mc))*vn1 Vc2=((2mn)/(mn+mc))*vn1 Physics Chapter 8 24 Center of mass The center of mass of a system of particles is a weighted average of the position of the particles If we have several particles with masses m1, m2, etc. and coordinates (x1 , y1), (x2 , y2), etc. The center of mass of the system is defined as the point with the following coordinates. Physics Chapter 8 25 Center of mass mx m x m x m x ... X m m m ... m i i 1 1 2 2 1 2 3 3 i 3 i i my m y m y m y ... Y m m m ... m i 1 1 2 1 2 2 3 3 i i 3 i i Physics Chapter 8 26 Example Find the center of mass of the Earth-Sun system. The distance between their centers is 1.49 x 1011 m. The mass of the Sun is 1.99 x 1030 kg. The mass of the Earth is 5.98 x 1024 kg. Physics Chapter 8 27 Example, continued mx m x m x m x ... X m m m ... m i i 1 1 2 2 1 1.9910 X 3 3 2 i 3 i i kg 0 m 5.981024 kg 1.491011 m 1.991030 kg 5.981024 kg 30 X 4.48105 m The radius of the sun is 6.95 x 108 m Physics Chapter 8 28 Velocity of the center of mass Vx mi vix m v m i iy Vy i mi i i i i V mi vi i M Where M = the total mass Physics Chapter 8 29 Momentum of the center of mass Total momentum equals the total mass times the velocity of the center of mass. P MV mi vi i Physics Chapter 8 30 Acceleration of the Center of Mass A mi ai i M Physics Chapter 8 31 Newton’s second law All internal forces between the particles cancel out Fext MA Physics Chapter 8 32 See page 196 The center of mass of the wrench moves in a straight line. The center of mass of the shell fragments follows the parabolic trajectory of the intact shell. Physics Chapter 8 33