CHAPTER 4. Impulse and momentum. CHAPTER s Objectives

Transcription

1 60 CHAPTER 4 Impulse and momentum CHAPTER s Objectives To understand the interaction between objects through the impulse and momentum concepts To introduce the law o conservation o momentum, and apply it to solve collision problems In Chapter we studied motion rom experimental point o view. In Chapter we associate motion with orce and developed a new deinition o orce a s an interaction between objects in contact or non contact. deinition o orce. The mechanism o this interaction in terms o work and energy was the ocus in chapter 3. In light o this, we now examine the collision process between two interacting objects using new concepts o impulse and momentum. To understand the two aspects o collision, the elastic and inelastic

2 6 4. Impulse J : Collision Problem Collision is deined as a short time interaction between two objects in contact, such as a tennis ball and a racket. Below is a brie description o this collision When the ball (object A) hits the racket (object B), they stay in contact or a short period o time t t t i. Typically, t is between to 0ms. The orces F A on B FB on A are action and reaction pair The orce is not constant, but varies with time. We use the term average orce to identiy this collision orce Impulse J is the product o the average orce and the time o contact and can be written as J F t (4.) Equation 4. implies the ollowing When the tennis ball got hit by the racket it receives an impulse Impulse is a vector quantity points always in the direction o the orce The stronger the orce acting, and the longer the time o contact, the greater the impulse will be The metric (SI) unit o impulse is Newton times second (N.s). Example 4. An average orce o 300 N acts or a time o 0.05 s on a gol ball. What is the magnitude o the impulse acting on the ball? Solution Given data: F 300N t 0.05s, J? J Ft J ( 300N) (0.05s) 5N. s 4. Momentum p The product o mass and velocity is called momentum p. It can be written as p mv (4.) Equation (4.) implies the ollowing

3 6 An object at rest has zero momentum The aster the object is moving or the more mass it has, the greater momentum it has. Momentum a vector quantity points in the direction o velocity. I the direction is not important, then we can drop the vector notation on momentum and speak o speed instead o velocity. The metric (SI) unit o momentum is a unit o mass times a unit o velocity (kg.m/s). Example 4. Find the momentum o 500 kg car traveling at 0m/s. Solution Given data: mass 500kg, wanted : p? v 0m / s p mv p (500kg) (0m / s) 5000kgm / s.50 4 kgm/ s The Relationship between Momentum and Inertia (mass) Momentum is related to both mass and speed. Thereore, it is much easier to stop a small object (bicycle, or example) than a truck both traveling at same speed. The truck, in this respect, has more inertia and is more diicult to stop or to move than the small object. Example 4.3 The drawing below shows a bowling ball o mass 5kg travels at m/s and a tennis ball with mass o 50g. Can both balls have the same momentum? I yes at what speed must the tennis ball travel to have the same momentum? Solution V =m/s m = 50g M = 5kg Given data: m 50g 0.50kg, M p BB p TB, Wanted : v( tennis ball) v 5kg, V m / s

4 63 MV mv v MV m (5kg)(m / s) 66m / s 0.50kg Thereore, the tennis ball must be moving with speed 66m/s to have same momentum as the bowling ball. 4.3 The Impulse-Momentum Theorem The impulse-momentum theorem states that: impulse changes momentum, or J p (4.3) Equation (4.3) can be rewritten as F t p p mv mv (4.4) i Equation (4.4) implies the ollowing means a change inalinitial i To increase momentum, i.e., p pi, increase F, or t, or both To decrease momentum, i.e., p pi, increase or decrease t Increasing Decreasing t less orce F t large orce F Decreasing and increasing the time o contact has a lot o applications. For example, reducing the time yield a large orce that breaks a stack o bricks shown in Figure 4.. The auto air bag is designed to protect the driver rom severe head damage during collision. The idea behind the bag is to increase the time o collision and thereore reduce the orce o collision to a minimum. (a) Figure 4.: (a) In a short time o impact, a large orce is produced to break a target. (b) In a long time o impact, a small orce is produced, which reduces head injuries o the driver and passenger. (b)

5 64 Example 4.4: A tennis ball bouncing o the loor. Examine the impulse-momentum change theorem by considering a tennis ball bouncing o the loor as shown in Figure 4. Δp p P i Floor Impulse Beore During Ater Figure 4. A typical collision problem o a tennis ball bouncing o a loor. When the ball hits the loor, the momentum decreases to zero as the ball comes to a temporary stop The ball receives an impulse J rom the loor in upward direction The ball gains momentum in opposite direction (upward) The ball leaves with p p pi p 0 p Example 4.5 A baseball o mass 0.5 kg has an initial velocity 0 0 m/s as it approaches a bat as shown in Figure 4.3. It is hit straight back to the right and leaves the bat with a inal velocity o +40 m/s. (a) Determine the 3 impulse applied to the ball by the bat. (b) Assume that the time o contact is.6 0 sec, ind the average orce exerted on the ball by the bat. (c) How much is the impulse exerted by the ball on the bat? Figure 4.3: A collision between a ball and bat. Credit: D. Knight Physics or scientists and engineers a strategic approach by Pearson-Addison Wesley 004.

6 65 (a) Apply the impulse-momentum theorem J mv mvi ( 0.5kg)(40m / s) (0.5kg)( 0m / s) 9kg m / s (b) Apply the equation that deines impulse J Ft F J t 9 kg m / s s 500N (c) Apply Newton s third law o action and reaction and get Example 4.6 Impulse exerted on the bat by the ball equals -9 kg m/s. The negative sign indicates a direction to the let o the origin o coordinate system. When you jump rom a height (say 3m) what kind o adjustment should you do to reduce sever injuries? Solution Bend your knees when landing. How you analyze the situation? Recall the equation F t p, rearrangeand p F t get Bending knees (say 5 cm) increases the time o impact with the ground and thereore decreases the orce F on the eet and legs. Landing on the ground with sti-legged (say with cm body move) decreases the time and increasing F. Typical exercise will show that the ground orce on the leg o a 70 kg person is 5 3 around. 0 N, while in bending the knee the orce on the same person is about 4.0 N. Clearly, the orce on the eet and legs is much less when the knees are bent. Indeed, the strength o the leg bone is not great enough to support the orce given above, so the leg would most likely break in such a sti landing. 4.4 Law o Conservation o Momentum The law o conservation o momentum is a direct result o Newton s third law. It applies when there are no net external orces acting on the objects involved. To show this, let us take an example o two balls collided a head-on collision as is shown in Figure 4.4. The only orces acting are internal orces.

7 66 Figure 4.4 A collision between two objects. By using the impulse=momentum theorem, we can write the orces acting on both balls as ollows F F on, on p t p t and F on By adding these two equations we get p p F on F on 0, or t t ( p p ) 0 t I the time change o the quantity ( p p ) 0, then this quantity is constant and doesn t change with time, or p p cons tan t (4.5) Equation (4.45)is a statement o a conservation law o momentum: i p p constant, then the sum o the momentum beore collision equals to the sum o momentum ater the collision. That is, mv i mv i mv mv (4.6) Example 4.7 A 0 g bullet is ired rom a 3 kg rile with speed o 500 m/s, as shown in Figure 4.5. What is (a) the initial momentum o the system (bullet and rile)? And (b) the recoil speed o the rile? Solution Given Data: m 0.00kg, m b ( v b ) r 3kg 500m / s, wanted : ( v ) r?

8 67 Figure 4.5: The rile and bullet. Credit: B.W. Tillery, E. D. Enger, and F. C. Ross, Integrated science, 3 rd Ed., McGraw Hill 004. ( a) p p b r ) ) i i p mb v m v bi r p ri bi ri (0.00g) 0 0 (3kg) The law o conservation o momentum states that the initial momentum o the system equals the inal momentum, or p p p b r p b r p p p b r bi p 0, ri or 0, (0.00kg) (500m / s) 5kgm / s The negative sign indicates that the rile s recoil is to the let with recoil velocity 5kgm/ s pr mr vr, vr.67m / s 3kg 4.5 Collisions The collision o two objects involves very large internal orces acting or very short period o time. To comprehend this topic, two dierent kinds o collision will be discussed here Elastic Collision Is a collision in which the total kinetic energy o the collided objects ater collision equals the total kinetic energy beore collision The collided object bounce a part and return to their original shape without a permanent deormation

9 Inelastic Collision Is a collision in which the total kinetic energy o the collided objects ater collision is not equal to the total kinetic energy beore collision. The two object experience a permanent deormation in their original shape In completely inelastic collision, the two objects coupled and move as a one object ater collision The kinetic energy is lost in two ways. It can be converted into heat because o riction.. It is spent in doing permanent deormation as in cars and part o it goes also as sound. In both elastic and inelastic collisions, the total momentum o both objects is conserved. Example 4.8 elastic collision Figure A ball o mass 0.6 kg traveling at 9 m/s to the right collides head on collision with a second ball o mass 0.3 kg traveling at 8 m/s to the let. Ater the collision, the heavier ball is traveling at.33 m/s to the let. What is the velocity o the lighter ball ater the collision? Figure 4.6: an elastic collision between two moving objects. Solution Given Data m 0.6kg, v wanted : v? 9m / s, m 0.3kg, v 8m / s, v.3m / s, m v m v m v m v v v mv m m 4,6m / s v m v (0.6kg)(9m / s) (0,3kg)( 8m / s) (0.6kg)(.3m / s) 0.3kg

10 69 Example 4.9 completely inelastic collision 4 Figure 4.7 shows a.750 kg railroad car traveling at 8 m/s to the east as shown in the drawing below is collided with another car o the same mass and initially at rest and couple with it. What is the velocity o the coupled system o cars ater the collision? Figure 4.7: an inelastic collision between two objects. Ater the collision the two objects coupled and move as one object with speed V. Solution Given Data 4 m m.750 kg, v 8m / s, v wanted : V? 0, m v m mv V m v m m ( m v m ) V mv 0 3.5m / s m m

11 70 SUMARRY OF CHPTER 4 Collision is a short time interaction between two objects. It involves very large and time dependent internal orces that dominate all external orces. Being action and reaction orces, these orces transer energy rom one object to another. Impulse J is the product o the interaction orce and the time o collision. It is a vector, which points always in the direction o the orce. Momentum is mass times velocity and is a measure o the object s motion. It is a vector, which always points in the direction o velocity. Momentum can be increased by either increasing the object s mass, or its speed,or both mass and speed. The impulse-momentum theorem states that the impulse equals the change in momentum. I the external orces acting on the two interacting objects are ignored, then the total momentum beore interaction equals the total momentum ater collision. This is the law o conservation o momentum; momentum beore equals momentum ater. Collision problems are classiied as either elastic collision or inelastic collision. In elastic collision both momentum and kinetic energy are conserved. In inelastic collision momentum only is conserved, where the object loses energy in the orm o heat and sound. Basic Equations Impulse is orce times time: J Favg t, or J Ft Momentum is mass times velocity: P mv, or P mv (4.) (4.) Basic Principles Impulse momentum theorem impulse equals change o momentum: J p, or Ft P P i (4.4) Conservation o momentum: momentum beore equals momentum ater: P i P, or m v m v m v m v (4.6)

12 7 Chapter 4 Worksheet Part: sentence completion. Impulse is times.. Momentum is times. 3. More inertia implies more. 4. equals change o momentum. 5. Energy equals energy. This is the law o. 6. Momentum equals momentum. This the law o. Part: Multiple choices. Impulse is A. A orce applied to an object. B. The initial orce applied to an object. C. The initial momentum applied to an object D. The change in momentum due to a orce being applied to an object during a short period o time.. Momentum is A. Equal to speed times weight. B. Equal to mass times velocity. C. Another alternative name to orce D. All o the above is correct. 3. What is the metric (SI) unit o momentum? A. Kg m/s. B. Newton. C. Kg m/s D. Newton. meter 4. Which o the ollowing statement is correctly expressing the conservation o total momentum o interacting objects? A. The total momentum always remains the same. B. The total momentum remains the same i there are no internal orces. C. The total momentum remains the same i there are no external orces. D. No one o the above is correct.

13 7 5. A 000 kg car is moving to the right at 30m/sec and collided with a wall and comes to rest at 0. sec. The average orce the car exerts on the wall is A.. x0 4 N to the right B. 3 x0 5 N to the let C. 6 x 0 4 N to the right D. None o the above 6. A 3kg object moves to the right with a speed o 4 m/sec. It collides in a perectly elastic collision with a 6kg object moving to the let at m/sec. What is the total kinetic energy ater the collision? A. 7 J B. 36 J C. 4 J D. 0 J 7. How long must a 00 N orce act to produce a change in momentum 00 kg.m/sec? A. 0.5 sec B sec C..0 sec D..0 sec 8. Which is a vector quantity? A. Energy B. Work C. Power D. Momentum 9. When the velocity o an object is doubled, its is also doubled A. Gravitational potential energy B. Acceleration C. Momentum D. Kinetic energy 0. Impulse is related to A. Kinetic energy B. Change in kinetic energy C. Momentum D. Change in momentum Part 3: True o alse. The impulse is always in the same directions as the average orce. A. True B. False

14 73. The momentum o an object remains the same when the net external orce acting on it is zero. A. True B. False 3. I the kinetic energy o an object is zero, then its momentum must not be zero. A. True B. False 4. A large orce always produces a larger impulse on a body than a smaller orce. A. True B. False 5. The kinetic energy is always conserved both in elastic collisions and inelastic collisions. A. True B. False 6. I two particles o dierent masses have equal kinetic energy they also have equal momentum. A. True B. False Part3: Exercises. An average orce o 50 N acts on a ball or 0.05 sec. (a) What is the magnitude o the impulse on the ball? (b) What is the change in the ball s momentum?. What is the momentum o 000 kg truck traveling with 5m/sec? 3. A ootball o mass.5 kg and a speed o m/sec and ping pong ball o mass 0.003kg and a speed o 4 m/sec. Which ball has the larger momentum? 4. A orce acts on a ball initially at rest or 0.05 sec. (a) What is the impulse on the ball? (b) What is the inal momentum o the ball? 5. A ather ice skater with a mass o 80kg pushes o against his child ice skater whose mass is 45kg. Both skaters were initially at rest. (a) What is the total momentum o both skaters ater they push o? (b) I the ather skater moves o with speed o.5 m/sec, what is the speed o the child?

15 74 6. A rocket ship, shown in the drawing below, at rest in space gives a short blast o its engine, iring 50kg o exhaust gas out the back end with an average velocity o 00 m/sec. What is the change in momentum o the rocket during this blast? 7. A rile and a bullet shown in the drawing below. The rile o mass o the rile.5 kg ires a bullet o 5g mass. The bullet moves with muzzle velocity v / = 500m/sec. (a) What is the momentum o the ired bullet? (b) I the external orces acting on the rile are ignored, what is its recoil velocity ater iring the bullet? 8. Consider the collision o train cars shown below as a perectly inelastic collision. ind the velocity o the coupled cars ater collision. Assume a 0,000 kg mass o each car Part4: Challenge exercises. A bullet is ired into a block o wood sitting on a block o ice. The bullet has an initial velocity o 350m/sec and a mass o 50g. The wooden block has a mass o kg and is initially ate rest. The bullet remains inside the wooden block ater collision. (a) assuming the momentum is conserved,

16 75 ind the velocity o wood and bullet ater the collision. (b) What is the magnitude o the impulse that acts on the block o wood?. A 000 kg ca is traveling north with speed o 30 m/sec collides head on with a 4000kg truck traveling south with speed o 0 m/sec. The car and truck stick together ater the collision. (a) What is the total momentum o the system o car and truck beore collision? (b) What is the velocity o both just ater the collision? (c) What is the total kinetic energies o the system beore collision? (d) What is the total kinetic energies just ater the collision? (e) Is the collision elastic or inelastic? Explain