Unit 5: Momentum

For this unit you must:1. Justify the selection of data needed to determine the relationship between the direction of the force

acting on an object and the change in momentum caused by that force2. Justify the selection of routines for the calculation of the relationships between changes in momentum of an

object, average force, impulse, and time of interaction3. Predict the change in momentum of an object from the average force exerted on the object and the interval of

time during which the force is exerted4. Analyze data to characterize the change in momentum of an object from the average force exerted on the

object and the interval of time during which the force is exerted5. Design a plan for collecting data to investigate the relationship between changes in momentum and average

force exerted on an object over time6. Calculate the change in linear momentum of a two-object system with constant mass in linear motion from a

representation of the system (data, graphs, etc.)7. Analyze data to find the change in linear momentum for a constant-mass system using the product of the mass

and the change in velocity of the center of mass8. Apply mathematical routines to calculate the change in momentum of a system by analyzing the average force

exerted over a certain time on the system9. Perform analysis on data presented as force-time graph and predict the change in momentum of a system10. Define open and closed systems for everyday situation and apply conservation concepts for energy, charge, and

linear momentum to those situations11. Make qualitative predictions about natural phenomena based on conservation of linear momentum and

restoration of kinetic energy in elastic collisions12. Apply the principles of conservation of momentum and restoration of kinetic energy to reconcile a situation that

appears to be isolated and elastic, but in which data indicate that linear momentum and kinetic energy are not the same after the interaction, by refining a scientific question to identify interactions that have not been considered. Students will be expected to solve qualitatively and/or quantitatively for one-dimensional situations and only qualitatively in two-dimensional situations.

13. Apply mathematical routines appropriately to problems involving elastic collisions in one dimension and justify the selection of those mathematical routines based on conservation of momentum and restoration of kinetic energy

14. Design an experimental test of an application of the principle of the conservation of linear momentum, predict an outcome of the experiment using the principle, analyze data generated by that experiment whose uncertainties are expressed numerically, and evaluate the match between the prediction and the outcome

15. Classify a given collision situation as elastic or inelastic, justify the selection of conservation of linear momentum and restoration of kinetic energy as the appropriate principles for analyzing an elastic collision, solve for missing variables, and calculate their values

16. Qualitatively predict, in terms of linear momentum and kinetic energy, how the outcome of a collision between two objects changes depending on whether the collision is elastic or inelastic

17. Plan data collection strategies to test the law of conservation of momentum in a two-object collision that is elastic or inelastic and analyze the resulting data graphically

18. Apply the conservation of linear momentum to a closed system of objects involved in an inelastic collision to predict the change in kinetic energy

19. Analyze data that verify conservation of momentum in collisions with and without an external friction force20. Classify a given collision situation as elastic or inelastic, justify the selection of conservation of linear momentum

as the appropriate solution method for an inelastic collision, recognize that there is a common final velocity for the colliding object in the totally inelastic case, solve for missing variables, and calculate their values

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21. Predict the velocity of the center of mass of a system when there is no interaction outside of the system but there is an interaction within the system (i.e., to recognize that interactions within a system do not affect the center of mass motion of the system and thus there is no external force)

Chapter 9: Linear Momentum and Collisions

Section 9-1 Linear Momentum

The change in linear momentum for a constant-mass system is the product of the mass of the system and the change in the velocity of the center of mass:

Example 1: A bean bag and a rubber ball of equal mass are dropped from the same height and hit a smooth surface. After striking the surface, is the change in momentum of the bean bag greater than, less than or equal to the change in momentum of the rubber ball? Justify your response.

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Example 2: At a city park, a person throws some bread into a duck pond. Two 4.00 kg ducks and a 9.00 kg goose paddle rapidly toward the bread as shown below. If the ducks swim 1.10 m/s and the goose swims with a speed of 1.30 m/s, find the total momentum of the three birds.

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Section 9-2 Momentum and Newton’s Second Law

Newton’s second law as is traditionally stated is only valid for systems with a constant mass:

Newton’s original statement of the second law involved a change in momentum and is valid even if the mass changes:

The above 2 statements are equivalent if the mass is constant, however the statement in terms of momentum is always valid.

Section 9-3 Impulse

The change in linear momentum of a system is given by the product of the average force on that system and the time interval during which the force is exerted:

The force that one object exerts on a second object changes the momentum of the second object The units for momentum are the same as the units of the area under the curve of force versus time graph. That

is to say the momentum is equal to the area under a force versus time graph Change in linear momentum and force are both vectors in the same direction The change in momentum of that object depends on the impulse, which is the product of the average force and

the time interval during which the interaction occurred:

Impulse-Momentum Theorem

Example 3: A ball of mass m = 0.25 kg rolling to the right at 1.3 m/s strikes a wall and rebounds to the left at 1.1 m/s. What is the impulse delivered to the ball by the wall?

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Example 4: A car traveling at 20 m/s crashes into a bridge abutment. Calculate the force on the 60.0 kg driver if the driver is stopped by:

a. A 20-m-long row of water-filled barrels.b. The crumple zone of her car (~1 m). Assume a constant acceleration.

Example 5: A person stands under an umbrella during a rain shower. A few minutes later the raindrop turn to hail – though the number of drops hitting the umbrella per time remains the same, as does their speed and mass. Is the force required to hold the umbrella in the hail greater than, less than, or equal to the force required in the rain? Justify your response.

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Example 6: During a storm, rain comes straight down with a velocity of -15 m/s and hits the roof of a car perpendicularly. The mass of rain per second that strikes the car rood is 0.060 kg/s.

a. Assuming that the rain comes to rest upon striking the car roof, find the average force exerted by the rain on the car roof.

b. Instead of rain, suppose hail is falling. The hail comes straight down at a mass rate of 0.060 kg/s and an initial velocity of -15 m/s. The hail strikes the roof perpendicularly, however, the hailstones bounce off the roof of the car. Would the force on the roof be smaller than, equal to, or greater than that of the rain?

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Example 7: After winning a prize on a game show, a 72 kg contestant jumps for joy. a. If the jump results in an upward speed of 2.1 m/s, what is the impulse experienced by the contestant?b. Before the jump, the floor exerts an upward force of mg on the contestant. What is the total upward force

exerted by the floor if the contestant pushes down on it for 0.36 s during the jump

Example 8: A rubber ball experiences the force shown as it bounces off the floor.a. What is the impulse on the ball?b. What is the average force on the ball?

Example 9: A 0.5 kg hockey puck slides to the right at 10 m/s. It is hit with a hockey stick that exerts the force shown. What is its approximate final speed?

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Example 10: A 150 g baseball is thrown with a speed of 20 m/s. It is hit straight back toward the pitcher at a speed of 40 m/s. The impulsive force of the bat on the ball has the shape shown in FIGURE 9.10. What is the maximum force Fmax that the bat exerts on the ball? What is the average force that the bat exerts on the ball?

Example 11: A force platform measures the horizontal force exerted on a person’s foot by the ground. The sketch shows force-versus-time data taken by a force platform for a 71 kg person who is jogging with increasing speed. The first part of the data indicates a negative force as the person’s foot comes to rest. The impulse during this time is -2.7 kg.m/s. The second part of the data shows a positive force as the person pushes off against the ground to gain speed. The impulse during this time period is 26.2 kg.m/s. For the time period from 0 to 0.80 s, what are

a. the average horizontal force exerted on the jogger, andb. the jogger’s change in speed?

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Section 9-4 Conservation of Linear Momentum

For all systems under all circumstances linear momentum is conservers. For an isolated or closed system momentum is constant. An open system is one that exchanges any conserved quantity with its surroundings.

Physics 1 includes a quantitative and qualitative treatment of conservation of momentum in one dimension and a semiquantitative treatment of conservation of momentum in two dimensions.

From Newton’s original statement of the second law we find that the change in momentum is given by:

If the net force acting on an object is zero, then the change in momentum is also zero. This is the statement of conservation of linear momentum:

Note: Momentum is a vector, therefore net force may be zero in one direction and not in another. In this case, momentum can be constant in one direction but change in another.

Conservation of Momentum for a System of Objects

internal forces occur in action-reaction pairs and are within the system the forces in action-reaction pairs are equal and opposite and hence must also sum to zero within the system thus, internal forces have no effect on the total momentum of a system if the net external force on a system is zero, its net momentum is conserved:

External forces can change the momentum of the system. The goal in problem solving is to choose the system such that momentum is conserved.

Example 12: Imagine two balls on a billiard table that is friction-free. Use the law of conservation of momentum to answer the following questions:

a. Just before the balls collide, suppose that a hole opens in the table beneath them. In this hypothetical situation, is the total momentum of the two-ball system conserved?

b. In a normal situation, when there is no hole in the table, is the momentum conserved?c. Answer part b. for a system that contains only one of the balls

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Example 13: Two ice skaters, Sandra and David, stand facing each other on frictionless ice. Sandra has a mass of 45 kg, David a mass of 80 kg. They then push off from each other. After the push, Sandra moves off at a speed of 2.2 m/s. What is David’s speed?

Example 14: Two groups of canoeists meet in the middle of a lake. After a brief visit, a person in canoe 1 pushes on canoe 2 with a force of 46 N to separate the canoes.

(A) If the mass of canoe 1 and its occupants is 130 kg, and the mass of canoe 2 and its occupants is 250 kg, find the momentum of each canoe after 1.20 s of pushing.

(B) Is the final kinetic energy of the system (i) greater than, (ii) less than, or (iii) equal to its initial kinetic energy?

Example 15: A 61 kg lumberjack stands on a 320 kg log that floats on a pong. Both the log and the lumberjack are at rest initially. In addition, the log points directly toward the shore, which we take to be the positive x direction.

(A) If the lumberjack trots along the log toward the shore with a speed v relative to the shore, the log moves away from the shore. Is the speed of the log relative to the shore greater than, less than, or equal to v? Explain.

(B) If the lumberjack trots with a velocity of 1.8 m/s relative to the shore, what is the velocity of the log relative to the shore?

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Example 16: Jack stands at rest on a skateboard. The mass of Jack and the skateboard together is 75 kg. Ryan throws a 3.0 kg ball horizontally to the right at 4.0 m/s to Jack, who catches it. What is the final speed of Jack and the skateboard?

Example 17: A 30 g ball is fired from a 1.2 kg spring-loaded toy rifle with a speed of 15 m/s. What is the recoil speed of the rifle?

Example 18: A freight train is being assembled in a switching yard and two boxcars are being coupled together. Car 1 has a mass of 65 000 kg and moves at a velocity of 0.80 m/s [E]. Car 2, with a mass of 92 000 kg and a velocity of 1.2 m/s [E], overtakes car 1 and couples to it. Neglecting friction, find the final velocity of the coupled cars.

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Section 9-5 Inelastic Collisions

In a collision between objects, linear momentum is conserved. In an inelastic collision, kinetic energy is not the same before and after the collision.

Kinetic energy usually decreases due to losses associated with sound, heat and deformation. It may increase if the collision sets off an explosion for instance.

When objects stick together after colliding, the collision is completely inelastic.

Example 19: A railroad car of mass m and speed v collides with and sticks to an identical railroad car that is initially at rest. After the collision, is the kinetic energy of the system (a) 1/2, (b) 1/3, or (c) ¼ of its initial kinetic energy?

Example 20: On a touchdown attempt, a 95.0 kg running back runs toward the end zone at 3.75 m/s. A 111 kg linebacker moving at 4.10 m/s meets the runner in a head-on collision. If the two players stick together, show that the collision is inelastic.

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Example 21: A 10.0 g bullet is fired into a 1.0 kg wood block, where it lodges. Subsequently the block slides 4.0 m across a floor (μk = 0.20 for wood on wood). What was the bullet’s speed?

Example 22: In a ballistic pendulum, an object of mass m is fired with an initial speed v, at the bob of a pendulum. The bob has a mass M, and is suspended by a rod of negligible mass. After the collision, the object and the bob stick together and swing through an arc, eventually gaining a height h. Find the height h in terms of m, M, v, and g.

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Example 23: A car with a mass of 950 kg and a speed of 16 m/s approaches an intersection, as shown. A 1300 kg minivan travelling at 21 m/s is heading for the same intersection. The car and minivan collide and stick together. Find the direction and speed of the wrecked vehicles just after the collision, assuming external forces can be ignored.

Section 9-6 Elastic Collisions

In a collision between objects, linear momentum is conserved. In an elastic collision, kinetic energy is the same before and after.

Most everyday collisions are poor approximations of being elastic on account of the energy being covert to other forms. Collisions whereby objects bounce off with little deformation are a reasonably good approximation. Elastic collisions are common in the subatomic world.

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Example 24: At an amusement park, a 98.5 kg bumper car moving with a speed of 1.18 m/s bounces elastically off a second 122 kg bumper car at rest. Find the final velocities of the two cars.

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Example 25: A ball of mass 0.250 kg and velocity +5.00 m/s collides head on with a ball of mass 0.800 kg that is initially at rest. No external forces act on the balls. If the collision is elastic, what are the velocities of the balls after the collision?

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If we consider object 1 moving to the right can colliding with object 2, initially at rest, we can combine our conservation of momentum equation:

With our conservation of kinetic energy equation:

To find the velocities of each object after collision:

Walker outlines a few special cases of this result.

(a) If m1 = m2

(b) If m1 << m2

(c) If m1 >> m2

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Example 26: A hoverfly is happily maintaining a fixed position about 3 m above the ground when an elephant charges out of the bush and collides with it. The fly bounces elastically off the forehead of the elephant. If the initial speed of the elephant is v, what is the final speed of the fly after collision?

Example 27: Two astronauts on opposite ends of a space ship are comparing lunches. One has an apple, the other has an orange. They decide to trade. Astronaut 1 tosses the 0.130 kg apple toward astronaut 2 with a speed of 1.11 m/s. The 0.160 kg orange is tosses from astronaut 2 to astronaut 1 with a speed of 1.21 m/s. Unfortunately, the fruits collide, sending the orange off with a speed of 1.16 m/s at an angle of 42.0° with respect to its original position. Find the final speed and direction of the apple, assuming an elastic collision. Give the apple’s direction relative to its original direction of motion.

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Section 9-7 Center of Mass

The velocity of the center of mass of the system cannot be changed by an interaction within the system Physics 1 includes no calculations of centers of mass Students are expected to be able to locate the center of mass of highly symmetric mass distributions, such as a

uniform rod or cube of uniform density, or two spheres of equal mass Where there is both a heavier and lighter mass, the center of mass is closer to the heavier mass (qualitative

treatment only)

The center of mass of a system depends upon the masses and positions of the object in the system. In an isolated system (a system with no external forces) the velocity of the center of mass does not change. When objects in a system collide, the velocity of the center of mass of the system will not change unless an external force is exerted on the system.

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