Linear Momentum Collisions and Energy Collisions in 2 Dimensions

Transcription

1 Linear Momentum Collisions and Energy Collisions in 2 Dimensions Lana Sheridan De Anza College Mar 1, 2019

2 Last time collisions elastic collision example inelastic collisions

3 Overview the ballistic pendulum 2 dimensional collisions

4 When is Momentum Conserved or Not: Example Billiard balls on a pool table. This is popular for modeling elastic collisions, since the collisions are very clean.

5 When is Momentum Conserved or Not: Example 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.

6 When is Momentum Conserved or Not: Example 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?

7 When is Momentum Conserved or Not: Example 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. The balls all roll toward the low side of the table.

8 262 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

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

10 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?

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

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

13 Collisions in 2 Dimensions The conservation of momentum equation is a vector equation. It will apply for any number of dimensions that are relevant in a question. p i = p f m 1 v 1i + m 2 v 2i = m 1 v 1f + m 2 v 2f

14 Collisions in 2 Dimensions The conservation of momentum equation is a vector equation. It will apply for any number of dimensions that are relevant in a question. p i = p f m 1 v 1i + m 2 v 2i = m 1 v 1f + m 2 v 2f In particular, we can write equations for each component of the momentum. In 2-d, with x and y components: x : m 1 v 1ix + m 2 v 2ix = m 1 v 1fx + m 2 v 2fx y : m 1 v 1iy + m 2 v 2iy = m 1 v 1fy + m 2 v 2fy

15 Collisions in 2 Dimensions The conservation of momentum equation is a vector equation. It will apply for any number of dimensions that are relevant in a question. p i = p f m 1 v 1i + m 2 v 2i = m 1 v 1f + m 2 v 2f In particular, we can write equations for each component of the momentum. In 2-d, with x and y components: x : m 1 v 1ix + m 2 v 2ix = m 1 v 1fx + m 2 v 2fx y : m 1 v 1iy + m 2 v 2iy = m 1 v 1fy + m 2 v 2fy If it is an elastic collision: K i = K f 1 2 m 1(v 1i ) m 2(v 2i ) 2 = 1 2 m 1(v 1f ) m 2(v 2f ) 2

16 S x m implies that th m important sub 1 Finalize This answer is not the maximum compression of the spring because iar example the two in m each other at the instant shown in Figure 9.10b. Can you determine 2 the surface. maximum For com su a for conservatio Collisions in 2 Dimensions As an example, consider the case of a glancing collision. a Before the collision S v 1i m 1 m 2 After the collision The velocity of particle 1 is in the x-direction. v 1f sin θ φ θ v2f sinφ S v 1f v 1f cos θ v 2f cos φ S v 2f v 1i After the collision 9.5 Collisions in Two S Dimension In Section 9.2, we v 1f sin showed θ that the momentum where the thr of served when the system is isolated. sent, For any respectiv collis implies that the momentum v 1f cos in θeach and of the the directi veloci θ important subset of collisions takes place Let in a us plane. consi iar example involving φ multiple collisions collides of object with p v 2f cos φ surface. For such two-dimensional collisions, (Fig. 9.11b), we pa ob for conservation of momentum: moves at an an v2f sinφ S v 2f sion. Applying m 1 1ix 1 m 2 vthat 2ix 5 the m 1 initial v 1fx 1 b m 1 v 1iy 1 m 2 v 2iy 5 m 1 v 1fy 1Dp Figure 9.11 An elastic, glancing where collision the three between subscripts two particles. on the velocity compo Dp sent, respectively, the identification of the object (1 and the velocity component (x, y). Let us consider a specific two-dimensional proble collides with particle 2 of mass m 2 initially at rest as i (Fig. 9.11b), particle 1 moves at an angle u with respec moves at an angle f with respect to the horizontal. T sion. Applying the law of conservation of momentum v 1f

17 S x m implies that th m important sub 1 Finalize This answer is not the maximum compression of the spring because iar example the two in m each other at the instant shown in Figure 9.10b. Can you determine 2 the surface. maximum For com su a for conservatio Collisions in 2 Dimensions As an example, consider the case of a glancing collision. a Before the collision S v 1i m 1 m 2 After the collision The velocity of particle 1 is in the x-direction. x-components: y-components: v 1f sin θ φ θ v2f sinφ S v 1f v 1f cos θ v 2f cos φ S v 2f v 1i After the collision 9.5 Collisions in Two S Dimension In Section 9.2, we v 1f sin showed θ that the momentum where the thr of served when the system is isolated. sent, For any respectiv collis implies that the momentum v 1f cos in θeach and of the the directi veloci θ important subset of collisions takes place Let in a us plane. consi iar example involving φ multiple collisions collides of object with p v 2f cos φ surface. For such two-dimensional collisions, (Fig. 9.11b), we pa ob for conservation of momentum: moves at an an v2f sinφ S v 2f sion. Applying m 1 1ix 1 m 2 vthat 2ix 5 the m 1 initial v 1fx 1 b m 1 v 1iy 1 m 2 v 2iy 5 m 1 v 1fy 1Dp Figure 9.11 An elastic, glancing where collision the three between subscripts two particles. on the velocity compo Dp sent, respectively, the identification of the object (1 and the velocity component (x, y). Let us consider a specific two-dimensional proble collides with particle 2 of mass m 2 initially at rest as i (Fig. 9.11b), particle 1 moves at an angle u with respec moves at an angle f with respect to the horizontal. T sion. Applying the law of conservation of momentum m 1 v 1i = m 1 v 1f cos θ + m 2 v 2f cos φ 0 = m 1 v 1f sin θ m 2 v 2f sin φ v 1f

18 Example Car collision ollisions A 1500 kg car traveling east with a speed of 25.0 m/s collides at an intersection with a 2500 kg truck traveling north at a speed of 20.0 m/s. Find the direction and magnitude of the velocity of the wreckage after the collision, assuming the vehicles stick together after the collision. ceptualize the situation before long the positive x direction and y S v f ediately before and immediately, we ignore the small effect that and model the two vehicles as an ore the vehicles sizes and model tic because the car and the truck 25.0i ˆ m/s u x 20.0j ˆ m/s ng momentum in the x direction initial momentum of the system the car. Similarly, the total initial f the truck. After the collision, let espect to the x axis with speed v f. 1 Serway & Jewett, page 265. Figure 9.12 (Example 9.8) An eastbound car colliding with a northbound truck.

19 Example Car collision This is an inelastic collision.

20 Example Exploding Rocket A rocket is fired vertically upward. At the instant it reaches an altitude of 1000 m and a speed of v i = 300 m/s, it explodes into three fragments having equal mass. One fragment moves upward with a speed of v 1 = 450 m/s following the explosion. The second fragment has a speed of v 2 = 240 m/s and is moving east right after the explosion. What is the velocity of the third fragment immediately after the explosion? (What is the sign of the change in kinetic energy of the system of the rocket parts?)

21 Example Exploding Rocket

22 Summary ballistic pendulum collisions in 2 dimensions 3rd Collected Homework will be posted. (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