Linear Momentum Inelastic Collisions

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1 Linear Momentum Inelastic Collisions Lana Sheridan De Anza College Nov 9, 2017

2 Last time collisions elastic collisions

3 Overview inelastic collisions the ballistic pendulum

4 Linear Momentum and Collisions Last time introduced two main kinds of collision: elastic inelastic

5 Linear Momentum and Collisions Last time introduced two main kinds of collision: elastic inelastic In elastic collisions both kinetic energy and momentum are conserved.

6 Linear Momentum and Collisions Last time introduced two main kinds of collision: elastic inelastic In elastic collisions both kinetic energy and momentum are conserved. In inelastic collisions only momentum is conserved, and some kinetic energy is lost to other forms of energy.

7 he collision. m v 1i v 2i sion, Perfectly we can 1 Inelastic Collisions ntum Special of the case of inelastic a collisions: (9.15) (9.14) m2 Before the collision, the particles move separately. After the S m vcollision, S 1i vthe 2i particles 1 move together. and S v a 2i (9.15) articles S v 1f and v After the collision, f the system particles m 1 move m 2 together. s tion S v 1i and in S v 2i b wo particles velocity! ties, S S v 1f and Figure 9.6 Schematic representation of a perfectly m 1 m 2 inelastic v f f the (9.16) system direction in head-on collision between two b (9.17) particles. Now the two particles stick together after colliding same final m2 p i = p f m 1 v 1i + m 2 v 2i = (m 1 + m 2 )v f

8 Perfectly Inelastic Collisions In this case it is straightforward to find an expression for the final velocity: So, m 1 v 1i + m 2 v 2i = (m 1 + m 2 )v f v f = m 1v 1i + m 2 v 2i m 1 + m 2

9 Question Quick Quiz 9.5 In a perfectly inelastic one-dimensional collision between two moving objects, what condition alone is necessary so that the final kinetic energy of the system is zero after the collision? (A) The objects must have initial momenta with the same magnitude but opposite directions. (B) The objects must have the same mass. (C) The objects must have the same initial velocity. (D) The objects must have the same initial speed, with velocity vectors in opposite directions. 1 Serway & Jewett, page 259.

10 Question Quick Quiz 9.5 In a perfectly inelastic one-dimensional collision between two moving objects, what condition alone is necessary so that the final kinetic energy of the system is zero after the collision? (A) The objects must have initial momenta with the same magnitude but opposite directions. (B) The objects must have the same mass. (C) The objects must have the same initial velocity. (D) The objects must have the same initial speed, with velocity vectors in opposite directions. 1 Serway & Jewett, page 259.

11 Perfectly Inelastic Collisions Perfectly inelastic collisions are the special case of inelastic collisions where the two colliding objects stick together. In this case the maximum amount of kinetic energy is lost. (The loss must be consistent with the conservation of momentum.

12 Perfectly Inelastic Collisions Perfectly inelastic collisions are the special case of inelastic collisions where the two colliding objects stick together. In this case the maximum amount of kinetic energy is lost. (The loss must be consistent with the conservation of momentum. Let s consider why.

13 Perfectly Inelastic Collisions Consider the same collision, viewed from different inertial frames. Suppose Alice sees: m 1 p A 1,i V 1,i m 2 m 1 m 2 V f In her frame, block 2 is at rest, and block 1 moves with velocity v 1,i. After the collision, both blocks move with velocity v f.

14 Perfectly Inelastic Collisions Consider the same collision, viewed from different inertial frames. Suppose Alice sees: m 1 p A 1,i V 1,i m 2 m 1 m 2 V f In her frame, block 2 is at rest, and block 1 moves with velocity v 1,i. After the collision, both blocks move with velocity v f. There is still some KE after the collision, but there must be at least some, since the momentum after cannot be zero.

15 Perfectly Inelastic Collisions Now consider what another observer, Bob, who is in the center-of-momentum frame, would see in the same collision: B B p1,i p2,i u 1,i u 2,i m1 m2 m1 m2 In his frame, both blocks are in motion, and p 1,i = p 2,i. After the collision, p 1,f + p 2,f = 0.

16 Perfectly Inelastic Collisions Now consider what another observer, Bob, who is in the center-of-momentum frame, would see in the same collision: B B p1,i p2,i u 1,i u 2,i m1 m2 m1 m2 In his frame, both blocks are in motion, and p 1,i = p 2,i. After the collision, p 1,f + p 2,f = 0. The final KE in this case is 0. (p 1,f = p 2,f = 0)

17 Perfectly Inelastic Collisions Observers in all inertial frames will see momentum conserved in any collision (with no external forces).

18 Perfectly Inelastic Collisions Observers in all inertial frames will see momentum conserved in any collision (with no external forces). For two colliding objects it is always possible to pick a frame where the total momentum is zero.

19 Perfectly Inelastic Collisions Observers in all inertial frames will see momentum conserved in any collision (with no external forces). For two colliding objects it is always possible to pick a frame where the total momentum is zero. If the objects stick together, the final kinetic energy in this frame is zero.

20 Perfectly Inelastic Collisions Observers in all inertial frames will see momentum conserved in any collision (with no external forces). For two colliding objects it is always possible to pick a frame where the total momentum is zero. If the objects stick together, the final kinetic energy in this frame is zero. No observer in another frame can assign less final kinetic energy than K = 1 2 mv 2 rel to the objects, where v rel is the relative speed of the other frame to the center-of-momentum frame.

21 Question Suppose Alice is in a frame such that she sees a 2 kg block moving at 3 m/s collide perfectly inelastically with a 1 kg block at rest. What is the change in kinetic energy observed by Alice in this collision? (A) 0 J (B) 3 J (C) 6 J (D) 9 J

22 Question Suppose Alice is in a frame such that she sees a 2 kg block moving at 3 m/s collide perfectly inelastically with a 1 kg block at rest. What is the change in kinetic energy observed by Alice in this collision? (A) 0 J (B) 3 J (C) 6 J (D) 9 J

23 Question Suppose Bob sees the same collision, but he is in a frame such that he sees the 2 kg block moving at 1 m/s collide perfectly inelastically with the 1 kg block moving in the opposite direction at 2 m/s. What is the change in kinetic energy observed by Bob? (A) 0 J (B) 3 J (C) 6 J (D) 9 J

24 Question Suppose Bob sees the same collision, but he is in a frame such that he sees the 2 kg block moving at 1 m/s collide perfectly inelastically with the 1 kg block moving in the opposite direction at 2 m/s. What is the change in kinetic energy observed by Bob? (A) 0 J (B) 3 J (C) 6 J (D) 9 J

25 Perfectly Inelastic Collisions Observers in different inertial frames will see different kinetic energies of the system. However, all inertial observers will see the same change in kinetic energy.

26 Perfectly Inelastic Collisions Observers in different inertial frames will see different kinetic energies of the system. However, all inertial observers will see the same change in kinetic energy. Since the kinetic energy cannot be negative, we there is a limit on how much can be lost: The loss cannot be more than the initial KE in the center-of-momentum frame.

27 Example sion? (b) What is the change in mechanical energy of the car truck system in the collision? (c) Account for this change in mechanical energy. 23. A 10.0-g bullet is fired into a stationary block of wood W having mass m kg. The bullet imbeds into the block. The speed of the bullet-plus-wood combination immediately after the collision is m/s. What was the original speed of the bullet? Page 285, # A car of mass m moving at a speed v 1 collides and couples with the back of a truck of mass 2m moving ini- S tially in the same direction as the car at a lower speed v 2. (a) What is the speed v f of the two vehicles immediately after the collision? (b) What is the change in kinetic energy of the car truck system in the collision? 25. A railroad car of mass kg is moving with a speed of 4.00 m/s. It collides and couples with three other coupled railroad cars, each of the same mass as the single car and moving in the same direction with an initial speed of 2.00 m/s. (a) What is the speed of the four cars after the collision? (b) How much mechanical energy is lost in the collision? 30. A S b p p b o su st g m sw 31. A AMT 1 M fa sl fr w 32. A S w su b

28 Example (a) Speed v f?

29 Example (a) Speed v f? v f = m 1v 1i + m 2 v 2i m 1 + m 2

30 Example (a) Speed v f? v f = m 1v 1i + m 2 v 2i m 1 + m 2 v f = mv 1 + 2mv 2 m + 2m v f = v 1 + 2v 2 3

31 Example (b) Change in kinetic energy? K = K f K i

32 Example (b) Change in kinetic energy? K = K f K i K = 1 ( 1 2 3mv f 2 2 mv ) 2 2mv 2 2 K = 1 ( ) 2 3m v1 + 2v 2 ( mv ) 2 2mv 2 2

33 Example (b) Change in kinetic energy? K = K f K i K = 1 ( 1 2 3mv f 2 2 mv ) 2 2mv 2 2 K = 1 ( ) 2 3m v1 + 2v 2 ( mv ) 2 2mv 2 2 K = m 6 ( (v v v 1 v 2 ) 3 ( v v 2 2 ))

34 Example (b) Change in kinetic energy? K = K f K i K = 1 ( 1 2 3mv f 2 2 mv ) 2 2mv 2 2 K = 1 ( ) 2 3m v1 + 2v 2 ( mv ) 2 2mv 2 2 K = m ( (v v v 1 v 2 ) 3 ( v v2 2 )) K = m ( 4v1 v 2 2v2 2 2v 2 ) 1 6 K = m ( 2v1 v 2 v2 2 v 2 ) 1 3

35 Example of A Collision System Billiard balls on a pool table. This is popular for modeling elastic collisions, since the collisions are very clean.

36 Example of A Collision System Billiard balls on a pool table. This is popular for modeling elastic collisions, since the collisions are very clean. The collisions are not perfectly elastic (you can hear a clack sound) but they are close to elastic. Momentum is conserved in these collisions on a flat table.

37 Example of A Collision System Billiard balls on a pool table. This is popular for modeling elastic collisions, since the collisions are very clean. The collisions are not perfectly elastic (you can hear a clack sound) but they are close to elastic. Momentum is conserved in these collisions on a flat table. What happens if one end of the table is propped up higher than the other?

38 Example of A Collision System Billiard balls on a pool table. This is popular for modeling elastic collisions, since the collisions are very clean. The collisions are not perfectly elastic (you can hear a clack sound) but they are close to elastic. Momentum is conserved in these collisions on a flat table. What happens if one end of the table is propped up higher than the other? An external net force acts; momentum is not conserved.

39 262 Chapter 9 Linear Momentum and Collisions Famous Example 9.6 continued The Ballistic Pendulum (Example 9.6) S S v1a vb m 1 m 2 a m 1 m 2 The ballistic pendulum is an apparatus used to measure the speed of a fast-moving projectile such as a bullet. A projectile of mass m 1 is fired into a large block of wood of mass m 2 suspended from some light wires. The projectile embeds in the block, and the entire system swings through a height h. How can we determine the speed of the projectile from a measurement of h? Figure 9.9 (Example 9.6) (a) Diagram of a ballistic pendulum. Notice that S v1a is the ve diately before the collision and S vb is the velocity of the projectile block system immediat tic collision. (b) Multiflash photograph of a ballistic pendulum used in the laboratory. Noting that v 2A 5 0, solve Equation 9.15 for v B : (1) v B 5 m 1v 1A m 1 1 m Categorize For the process during which the projectile block combination swings configuration we ll call C), we focus on a different system, that of the projectile, the bl 1 Serway this & part Jewett, of the page problem 262. as one involving an isolated system for energy with no noncons h Cengage Learning/Charles D. Winters b

40 The Ballistic Pendulum We know m 1, m 2, and h. We want to know the speed of the bullet, v 1.

41 The Ballistic Pendulum We know m 1, m 2, and h. We want to know the speed of the bullet, v 1. Step 1: how does the speed of the block v b depend on the bullet speed?

42 The Ballistic Pendulum We know m 1, m 2, and h. We want to know the speed of the bullet, v 1. Step 1: how does the speed of the block v b depend on the bullet speed? Conservation of momentum, perfectly inelastic collision: (m 1 + m 2 )v b = m 1 v 1 + m 2 (0) v b = m 1 v 1 m 1 + m 2

43 The Ballistic Pendulum Step 2: What happens after the bullet hits the block?

44 The Ballistic Pendulum Step 2: What happens after the bullet hits the block? How does v b relate to h?

45 The Ballistic Pendulum Step 2: What happens after the bullet hits the block? How does v b relate to h? Conservation of energy: K + U g = 0 (0 1 2 (m 1 + m 2 )v 2 b ) + ((m 1 + m 2 )gh 0) = (m 1 + m 2 )v 2 b = (m 1 + m 2 )gh

46 The Ballistic Pendulum Step 2: What happens after the bullet hits the block? How does v b relate to h? Conservation of energy: K + U g = 0 (0 1 2 (m 1 + m 2 )v 2 b ) + ((m 1 + m 2 )gh 0) = (m 1 + m 2 )vb 2 = (m 1 + m 2 )gh Replace v b = m 1v 1 m 1 +m 2 : ( ) 1 2 (m m1 v m 2 ) = (m 1 + m 2 )gh m 1 + m 2 ( m 2 1 v1 2 ) = 2(m 1 + m 2 )gh m 1 + m 2 ( ) m1 + m 2 2gh v 1 = m 1

47 Summary inelastic collisions ballistic pendulum 3rd Collected Homework! due Monday, Nov 20. (Uncollected) Homework Serway & Jewett, Look at example 9.9 on page 266. PREV: Ch 9, onward from page 285. Probs: 23, 25, 27, 29, 31 Ch 9, onward from page 275. Probs: 35, 37, 41, 43

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