PHYSICS 231 INTRODUCTORY PHYSICS I

Transcription

1 PHYSICS 231 INTRODUCTORY PHYSICS I Lecture 8

2 Last Lecture Work for nonconstant force F x Spring force F =!kx x Potential Energy of Spring PE = 1 2 kx2 Power P = "W "t P = Fv = "KE "t

3 Chapter 6 Momentum and Collisions

4 Momentum Definition: r p = m r v Newton s 2 nd Law: " r F = m "r v "t r F = "r p "t

5 Conservation of Momentum True for isolated particles (no external forces) Proof: Recall F 12 =-F 21, (Newton s 3 rd Law) r F 12 + F r 21 = 0 r " #r p 1 #t + # #t r " p i p 2 = 0 " # p r 1 + # p r 2 = 0 r p 1f + p r 2 f = p r 1i + p r 2i for isolated particles never changes!

6 Momentum is a Vector quantity Both!p x and!p y are conserved p x = mv x p y = mv y

7 Example 6.1 An astronaut of mass 80 kg pushes away from a space station by throwing a 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? m/s

8 Center of mass does not accelerate X cm! m 1x 1 + m 2 x 2 + m 3 x (m 1 + m 2 + m )!X cm = m 1!x 1 + m 2!x 2 + m 3!x (m 1 + m 2 + m ) =!t " m 1(!x 1 /!t) + m 2 (!x 2 /!t) + m 3 (!x 3 /!t) +... (m 1 + m 2 + m ) =!t " p 1 + p 2 + p (m 1 + m 2 + m ) = 0 if totalp iszero

9 Example 6.2 Ted and his ice-boat (combined mass = 240 kg) rest on the frictionless surface of a frozen lake. A heavy rope (mass of 80 kg and length of 100 m) is laid out in a line along the top of the lake. Initially, Ted and the rope are at rest. At time t=0, Ted turns on a wench which winds 0.5 m of rope onto the boat every second. a) What is Ted s velocity just after the wench turns on? 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? 12.5 m f) How far did the center-of-mass of the rope move? m

10 Example 6.3 A 1967 Corvette of mass 1450 kg moving with a velocity of 100 mph (= 44.7 m/s) slides on a slick street and collides with a Hummer of mass 3250 kg which is parked on the side of the street. The two vehicles interlock and slide off together. What is the speed of the two vehicles immediately after they join? 13.8 m/s =30.9 mph

11 Impulse = F!t =!p Impulse Useful for sudden changes where the exact details of the force are difficult to determine For nonconstant F, Impulse = Area under F vs. t curve

12 Bungee Jumper Demo

13 Example 6.4 A pitcher throws a 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 2.0 x 10 3 s. c) What is the acceleration experienced by the ball? a) kg"m/s b) 6,525 N c) 45,000 m/s 2

14 Collisions Momentum is always conserved in a collision Classification of collisions: ELASTIC Both energy & momentum are conserved INELASTIC Momentum conserved, not energy Perfectly inelastic -> objects stick Lost energy goes to heat

15 Examples of Perfectly Inelastic Collisions Catching a baseball Football tackle Cars colliding and sticking Bat eating an insect Examples of Perfectly Elastic Collisions Superball bouncing Electron scattering

16 Ball Bounce Demo

17 Example 6.5a 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 and the net momentum are conserve D) Neither are conserved

18 Example 6.5b A astronaut floating in space catches a baseball A) Momentum of the astronaut+baseball is conserved B) Mechanical energy of the astronaut+baseball is conserve C) Both mechanical energy and momentum are conserved D) Neither are conserved

19 Example 6.5c 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 and momentum are conserved D) Neither are conserved

20 Perfectly Inelastic collision in 1-dimension m 1 v 1i + m 2 v 2i = ( m 1 + m 2 )v f Final velocities are the same

21 Example 6.6 A 5879-lb (2665 kg) Cadillac Escalade going 35 mph =smashes into a 2342-lb (1061 kg) Honda Civic also moving at 35 mph=15.64 m/s in the opposite direction.the cars collide and stick. a) What is the final velocity of the two vehicles? b) What are the equivalent brick-wall speeds for each vehicle? a) 6.73 m/s = 15.1 mph b) 19.9 mph for Cadillac, 50.1 mph for Civic

22 Example 6.7 A proton (m p =1.67x10-27 kg) elastically collides with a target proton which then moves straight forward. If the initial velocity of the projectile proton is 3.0x10 6 m/s, and the target proton bounces forward, what are a) the final velocity of the projectile proton? b) the final velocity of the target proton? x10 6 m/s

23 Elastic collision in 1-dimension 1. Conservation of Energy: 1 m 2 1v 2 1i + 1 m 2 2v 2 2i = 1 m 2 1v 2 1 f + 1 m 2 2 2v 2 f 2. Conservation of Momentum: m 1 v 1i + m 2 v 2i = m 1 v 1 f + m 2 v 2 f (2) Rearrange both equations and divide: m 1 v 2 2 1i " v 1 f (1) 2 2 ( ) = m 2 ( v 2 f " v 2i ) (1) ( )( v 1i + v 1 f ) = m 2 ( v 2 f " v 2i )( v 2 f + v 2i ) ( ) = m 2 ( v 2 f " v 2i ) (2) m 1 v 1i " v 1 f m 1 v 1i " v 1 f v 1i + v 1 f = v 2 f + v 2i " v 1i # v 2i = #( v 1 f # v ) 2 f

24 Elastic collision in 1-dimension Final equations for head-on elastic collision: m 1 v 1i + m 2 v 2i = m 1 v 1 f + m 2 v 2 f v 1i " v 2i = "( v 1f " v ) 2 f Relative velocity changes sign Equivalent to Conservation of Energy

25 Example 6.8 An proton (m p =1.67x10-27 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 3.0x10 6 m/s, and 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 =-1.0x10 6 m/s v d = 2.0x10 6 m/s Head-on collisions with heavier objects always lead to reflections

26 Example 6.9a The mass M 1 enters from the left with velocity v 0 and strikes the mass M 2 =M 1 which is initially at rest. The collision is perfectly elastic. a) Just after the collision v 2 v 0. A) > B) < C) =

27 Example 6.9b The mass M 1 enters from the left with velocity v 0 and strikes the mass M 2 =M 1 which is initially at rest. The collision is perfectly elastic. Just after the collision v 1 0. A) > B) < C) =

28 Example 6.9c The mass M 1 enters from the left with velocity v 0 and strikes the mass M 2 =M 1 which is initially at rest. The collision is perfectly elastic. Just after the collision P 2 M 1 v 0. A) > B) < C) =

29 Example 6.9d The mass M 1 enters from the left with velocity v 0 and strikes the mass M 2 =M 1 which is initially at rest. The collision is perfectly elastic. At maximum compression, the energy stored in the spring is (1/2)M 1 v 0 2 A) > B) < C) =

30 Example 6.9e The mass M 1 enters from the left with velocity v 0 and strikes the mass M 2 <M 1 which is initially at rest. The collision is perfectly elastic. Just after the collision v 2 v 0. A) > B) < C) =

31 Example 6.9f The mass M 1 enters from the left with velocity v 0 and strikes the mass M 2 <M 1 which is initially at rest. The collision is perfectly elastic. Just after the collision v 1 0. A) > B) < C) =

32 Example 6.9g The mass M 1 enters from the left with velocity v 0 and strikes the mass M 2 <M 1 which is initially at rest. The collision is perfectly elastic. Just after the collision P 2 M 1 v 0. A) > B) < C) =

33 Example 6.9h The mass M 1 enters from the left with velocity v 0 and strikes the mass M 2 <M 1 which is initially at rest. The collision is perfectly elastic. At maximum compression, the energy stored in the spring is (1/2)M 1 v 0 2 A) > B) < C) =

34 Example 6.9i The mass M 1 enters from the left with velocity v 0 and strikes the mass M 2 >M 1 which is initially at rest. The collision is perfectly elastic. Just after the collision v 2 v 0. A) > B) < C) =

35 Example 6.9j The mass M 1 enters from the left with velocity v 0 and strikes the mass M 2 >M 1 which is initially at rest. The collision is perfectly elastic. Just after the collision v 1 0. A) > B) < C) =

36 Example 6.9k The mass M 1 enters from the left with velocity v 0 and strikes the mass M 2 >M 1 which is initially at rest. The collision is perfectly elastic. Just after the collision P 2 M 1 v 0. A) > B) < C) =

37 Example 6.9l The mass M 1 enters from the left with velocity v 0 and strikes the mass M 2 >M 1 which is initially at rest. The collision is perfectly elastic. At maximum compression, the energy stored in the spring is (1/2)M 1 v 0 2 A) > B) < C) =