An astronaut of mass 80 kg pushes away from a space Both!p x

Transcription

1 Chapter 6 Momentum Collisions Definition: Momentum Important because it is CONSERVED proof: p = m v F = m v t = p t Ft = p Since F 12 =-F 21, p 1 + p 2 = 0 p i for isolated particles never changes Vector quantity p x = mv x p y = mv y Example 61 An astronaut of mass 80 kg pushes away from a space Both p x p y are conserved station by throwing a 075- kg wrench which moves with a velocity of 24 m/s relative to the original frame of the astronaut What is the astronaut s recoil speed? 0225 m/s Center of mass does not accelerate X cm m 1x 1 + m 2 x 2 + m 3 x 3 + (m 1 + m 2 + m 3 + ) Do the back-to-back demo X cm = m 1x 1 + m 2 x 2 + m 3 x 3 + (m 1 + m 2 + m 3 + ) = t " m 1(x 1 / t) + m 2 (x 2 / t) + m 3 (x 3 / t) + (m 1 + m 2 + m 3 + ) p = t " 1 + p 2 + p 3 + (m 1 + m 2 + m 3 + ) = 0 if totalp iszero

2 Example 62 Ted his ice-boat (combined mass = 240 kg) rest on the frictionless surface of a frozen lake A heavy rope (mass of 80 kg length of 100 m) is laid out in a line along the top of the lake Initially, Ted the rope are at rest At time t=0, Ted turns on a winch which winds 05 m of rope onto the boat every second a) What is Ted s velocity just after the winch turns on? 0125 m/s b) What is the velocity of the rope at the same time? m/s c) What is the Ted s speed just as the rope finishes? 0 d) How far did the center-of-mass of Ted+boat+rope move 0 e) How far did Ted move? 125 m f) How far did the center-of-mass of the rope move? -375 m Example 63 A 1967 Corvette of mass 1450 kg moving with a velocity of 100 mph (= 447 m/s) slides on a slick street collides with a Hummer of mass 3250 kg which is parked on the side of the street The two vehicles interlock slide off together What is the speed of the two vehicles immediately after they join? 138 m/s =309 mph Impulse = Ft = p Impulse Bunjee Jumper Demo Useful for sudden changes when not interested in force but only effects of force Graphical Representation of Impulse Impulse = Ft = p For complicated force, "p is area under F vs t curve F Total Impulse F Example 64 A pitcher throws a 0145-kg baseball so that it crosses home plate horizontally with a speed of 40 m/s It is hit straight back at the pitcher with a final speed of 50 m/s a) What is the impulse delivered to the ball? b) Find the average force exerted by the bat on the ball if the two are in contact for 20 x 10 3 s "t t c) What is the acceleration experienced by the ball? a) 1305 kg#m/s b) 6,525 N c) 45,000 m/s 2

3 Elastic & Inelastic Collisions ELASTIC: Both energy & momentum are conserved INELASTIC Momentum conserved, not energy Perfectly inelastic -> objects stick Lost energy goes to heat Examples of Inelastic Collisions Catching a baseball Football tackle Cars colliding sticking Bat eating an insect Examples of Perfectly Elastic Collisions Superball bouncing Electron scattering Ball Bounce Demo Example 65a A superball bounces off the floor, A) The net momentum of the earth+superball is conserved B) The net energy of the earth+superball is conserved C) Both the net energy the net momentum are conserve D) Neither are conserved Example 65b An astronaut floating in space catches a baseball A) Momentum of the astronaut+baseball is conserved B) Mechanical energy of the astronaut+baseball is conserved C) Both mechanical energy momentum are conserved D) Neither are conserved Example 65c A proton scatters off another proton No new particles are created A) Net momentum of two protons is conserved B) Net kinetic energy of two protons is conserved C) Both kinetic energy momentum are conserved D) Neither are conserved

4 Example 66 A 5879-lb (2665 kg) Cadillac Escalade going 35 mph smashes into a 2342-lb (1061 kg) Honda Civic also moving at 35 mph=1564 m/s in the opposite directionthe cars collide stick a) What is the final velocity of the two vehicles? b) What are the equivalent brick-wall speeds for each vehicle? a) 673 m/s = 151 mph b) 199 mph for Cadillac, 501 mph for Civic Example 67 A proton (m p =167x10-27 kg) elastically collides with a target proton which then moves straight forward If the initial velocity of the projectile proton is 30x10 6 m/s, the target proton bounces forward, what are a) the final velocity of the projectile proton? b) the final velocity of the target proton? 00 30x10 6 m/s Work in center-of-mass frame (elastic) 1 Find com velocity subtract it from both v 1 v 2 v com = m 1 v 1,i + m 2 v 2,i m 1 + m 2 Note: v 1,i " v 1,i # v com v 2,i " v 2,i # v com m 1 v 1,i + m 2 v 2,i = 0 2 Problem is easy to solve in this frame: Elastic: v 1,i = "v 1,i, v 2,i = "v 2,i Working out answer in center-of-mass 3 Relative velocity switches sign v 1, f v 2, f = (v 2,i v 1,i ) For elastic head-on collisions: m 1 v 1, f + m 2 v 2, f = m 1 v 1,i + m 2 v 2,i v 1, f v 2, f = v 2,i v 1,i Equivalent to energy conservation Example 68 An proton (m p =167x10-27 kg) elastically collides with a target deuteron (m D =2m p ) which then moves straight forward If the initial velocity of the projectile proton is 30x10 6 m/s, the target deuteron bounces forward, what are a) the final velocity of the projectile proton? b) the final velocity of the target deuteron? v p =-10x10 6 m/s v d = 20x10 6 m/s Head-on collisions with heavier objects lead to reflections Example 69a =M 1 which is initially at rest The a) Just after the collision v 2

5 Example 69b =M 1 which is initially at rest The Just after the collision v 1 0 Example 69c =M 1 which is initially at rest The Just after the collision P 2 M 1 Example 69d =M 1 which is initially at rest The At maximum compression, the energy stored in the spring is (1/2)M 1 2 Example 69e <M 1 which is initially at rest The Just after the collision v 2 Example 69f <M 1 which is initially at rest The Just after the collision v 1 0 Example 69g <M 1 which is initially at rest The Just after the collision P 2 M 1

6 Example 69h <M 1 which is initially at rest The At maximum compression, the energy stored in the spring is (1/2)M 1 2 Example 69i >M 1 which is initially at rest The Just after the collision v 2 Example 69j >M 1 which is initially at rest The Example 69k >M 1 which is initially at rest The Just after the collision v 1 0 Just after the collision P 2 M 1 Example 69l >M 1 which is initially at rest The Example 610 At maximum compression, the energy stored in the spring is (1/2)M 1 2 Ballistic Pendulum: used to measure speed of bullet 05-kg block of wood swings up by height h = 65 cm after stopping 80-g bullet 227 m/s

7 Example 611 A 5-g bullet traveling at 500 m/s embeds in a 1495 kg block of wood resting on the edge of a 09-m high table How far does the block l from the edge of the table? Example 612 Tarzan (M=80 kg) swings on a 12- m vine by letting go from an angle of 60 degrees from the vertical At the bottom of his swing, he picks up Jane (m=50 kg) To what angle do Tarzan Jane swing? 358 degrees (indepedent of L or g) 714 cm Example 612a Tarzan (M=80 kg) swings on a 12- m vine by letting go from an angle of 60 degrees from the vertical At the bottom of his swing, he picks up Jane (m=50 kg) To what angle do Tarzan Jane swing? To calculate Tarzan s speed just before he picks up Jane, you should apply: A) Conservation of Energy B) Conservation of Momentum Example 612b Tarzan (M=80 kg) swings on a 12- m vine by letting go from an angle of 60 degrees from the vertical At the bottom of his swing, he picks up Jane (m=50 kg) To what angle do Tarzan Jane swing? To calculate Tarzan&Jane s speed just after their collision (given the previous answer) you should apply: A) Conservation of Energy B) Conservation of Momentum Example 612c Tarzan (M=80 kg) swings on a 12- m vine by letting go from an angle of 60 degrees from the vertical At the bottom of his swing, he picks up Jane (m=50 kg) To what angle do Tarzan Jane swing? To calculate Tarzan&Jane s final height (given the previous answer) you should apply: (Perfectly) Inelastic Collisions in Two Dimensions SKIP LAST 4 SLIDES Two Equations : Two unknowns (v fx, v fy ) m 1 v 1ix + m 2 v 2ix = (m 1 + m 2 )v fx m 1 v 1iy + m 2 v 2iy = (m 1 + m 2 )v fy A) Conservation of Energy B) Conservation of Momentum

8 Elastic Collisions in Two Dimensions Example 614 Three Equations : Four Unknowns (v 1f,v 2f,$ 1f,$ 2f ) 1 2 mv 1i mv 2i 2 = 1 2 mv 1 2 f mv 2 2 f m 1 v 1ix + m 2 v 2ix = m 1 v 1 fx + m 2 v 2 fx m 1 v 1iy + m 2 v 2iy = m 1 v 1 fy + m 2 v 2 fy A 1200-kg vehicle moving at 250 m/s east collides with a vehicle of mass 1500 kg moving northward at 200 m/s After they join, what is their final speed direction? = 0 v f = 157 m/s $ f = 45 Example 615 A projectile proton moving with =120,000 m/s collides elastically with a second proton at rest If one proton leaves at an angle $=30, a) what is its speed? b) what are the speed direction of the second proton? a) v 1 = 103,923 m/s b) v 2 = 60,000 m/s $ 2 = 60