Assume the ball has the same speed when it leaves the pitcher's hand and ..... over each bump, meaning that the cart loses a little momentum and a lit...

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AP PHYSICS 1

Momentum

2016 EDITION $MJDLPOUIFGPMMPXJOHMJOLPSTDBOUIF23DPEF UPDPNQMFUFUIFFWBMVBUJPOGPSUIF4UVEZ4FTTJPO IUUQTXXXTVSWFZNPOLFZDPNS[email protected]

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Pre-Assessment Questions P1. A heavy bus and a light car both start at rest at the starting line of a long race track. At t = 0, the race begins. Both vehicles have the same constant net force during the race. After 10 seconds, the car passes the finish line. (a) Which vehicle crosses the finish line with more kinetic energy? Explain your reasoning.

(b) Which vehicle crosses the finish line with more momentum? Explain your reasoning.

(c) At time t = 5 seconds (while the race is still happening), which vehicle has more kinetic energy? Explain.

(d) At time t = 5 seconds (while the race is still happening), which vehicle has more momentum? Explain.

P2: A pitcher throws a baseball and a catcher catches it. Assume the ball has the same speed when it leaves the pitcher’s hand and when it reaches the catcher’s mitt. (a) Who exerts a greater magnitude force on the ball? _____ Pitcher

_____ Catcher

_____ Both the same

_____ Not enough information to tell

(b) Who exerts a greater magnitude impulse on the ball? _____ Pitcher

_____ Catcher

_____ Both the same

_____ Not enough information to tell

(c) Explain your answers

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Momentum Basics:

! An object has momentum if it has mass and is moving. The symbol for momentum is p . The equation for the ! ! momentum of an object is given by the equation p = mv . Note that momentum is a vector, and that an object’s momentum has the same direction as its velocity. The direction of momentum can be indicated by using a positive value for rightward or upward motion and a negative value for leftward or downward motion. An object’s momentum can change when a net force acts on the object for an interval of time. The action of a net force during an interval of time is called impulse. • •

! If the net force acting on the object is constant during the interval it acts, impulse is given by Fnet Δt . If the net force acting on the object is changing during the interval, impulse is given by the area under a net force vs. time graph.

The Impulse-Momentum Theorem states that the impulse applied to an object is equal to the object’s change in ! ! momentum. On the AP Physics 1 Table of Equations, this appears as the equation Δp = Fnet Δt . The units for momentum are either kg•m/s or N•s. Both units are equivalent, so 24 kg•m/s of momentum is the same as 24 N•s of momentum. Here are examples of objects that have this amount of momentum: • • •

24 kg•m/s of momentum A 4 kg object moving at 6 m/s An 8 kg object moving at 3 m/s A 2 kg object moving at 12 m/s

• • •

24 N•s of momentum An object accelerated from rest by a 4 N force acting for 6 seconds An object accelerated from rest by a 8 N force acting for 3 seconds An object accelerated from rest by a 2 N force acting for 12 seconds

Conservation of Momentum Momentum is a quantity that is transferred from one object or system to another object or system by the action of a force. Consider Newton’s Third Law: If a force acts on an object, then an equal force acts on another object in the opposite direction. This means that, if a rightward force gives Object A rightward momentum (or takes away leftward momentum), then some other Object B somewhere has a leftward force that is giving Object B leftward momentum (or taking away rightward momentum). If Object A and Object B are part of the same system, then the forces that the two objects exert on each other are internal to the system. All forces are due to an object and act on another object. For example, your weight is the force acting on you due to the Earth. If you and the Earth were part of the same system, that weight force would be internal to the you-Earth system. If the Earth were not part of the system, then the weight force would be considered an external force, because the weight force acting on you is due to an object that is not part of the system. The Law of Conservation of Momentum states: If no external forces act on a system, then the total momentum of the entire system remains constant even as the objects within a system exert forces on each other. We normally apply Conservation of Momentum to “collisions” (two objects come together and push on each other) and “explosions” (two objects that are initially together push each other apart). The Conservation of Momentum equation (for objects 1 and 2) could be written as:

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m1v1i + m2 v 2i = m1v1 f + m2 v 2 f Types of Collisions Collisions are classified into four types. All collisions conserve momentum, but the total kinetic energy of all objects in a system may not be conserved during a collision. Kinetic energy could be transformed into other forms during the collision, or kinetic energy could actually be added to a system during a collision. Type

Energy Comparison

Word Description

Explosion

Ki < Kf

Some potential energy is released during the collision and the result is that the system has MORE kinetic energy after the collision than before.

Elastic

Ki = Kf

No kinetic energy is transferred to or from other forms during the collision.

Inelastic

Ki > Kf

Some kinetic energy is transferred to other forms such as (heat or sound) during the collision.

Perfectly Inelastic

Ki > Kf

Lots of kinetic energy is transferred to other forms during the collision, AND the objects stick together as a result of the collision.

An extremely dangerous and wrong thing that students believe is that “if the objects stick together, then the collision is inelastic, but if they don’t stick together, the collision is elastic”. Be warned! It is possible to have an inelastic collision (some KE is lost to other forms) and yet the objects don’t stick together. Any time a collision takes place and you can hear a sound from it, that collision is inelastic since some KE was used to make the sound. The Relationship Between Momentum and Center of Mass All systems have a special location called the system’s “center of mass” (CM). The CM of a system is closer to the heavier objects in the system. If the objects in the system move, the CM can move. If any objects in the system accelerate, the CM can also accelerate. Two things to know about the motion of the center of mass:

the velocity of the system' s center of mass =

the total momentum of the system the total mass of the system

the acceleration of the system' s center of mass =

the net external force on the system the total mass of the system

The top equation comes from p = mv, dividing both sides by m, and taking p and m to be “total amounts” for the system. The bottom equation comes from F = ma, dividing both sides by m, and taking F and m to be “total amounts” for the system. If there are no external forces on the system, then the total momentum of the system remains the same. This means that the velocity of the system’s center of mass cannot change, and the system’s center of mass cannot accelerate. Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org

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Multiple-Choice Questions M1. Which of the following statements is correct? (A) An object can have zero kinetic energy and nonzero momentum. (B) An object can have zero momentum and nonzero kinetic energy. (C) A system of objects can have zero total kinetic energy and non-zero total momentum. (D) A system of objects can have zero total momentum and non-zero total kinetic energy.

Science

Questions 4–5: A 1 kg mass moving 8 m/s to the right collides with a 4 kg mass at rest, as shown in the top diagram. After the collision, the 4 kg mass has the velocity shown in the bottom diagram. The arrows are drawn to-scale against a grid so that their components can be related visually.

M2. A force F applied to an object of mass m causes it to move in a straight line a distance D during a time interval T. The object gains a momentum p during this interval. In which of the following cases will the object gain the same momentum p? (A) A force F is applied to a mass 2m during a time interval T. (B) A force F is applied to a mass 2m as it travels through a distance D. (C) A force 2F is applied to a mass 2m during a time interval T. (D) A force 2F is applied to a mass 2m as it travels through a distance D.

M4. Which vector in the bottom diagram represents the velocity of the 1 kg mass after the collision? M5. Which vector shown in the bottom diagram has the same direction as the impulse applied to the 1 kg mass as a result of the collision?

M3. A block of mass m is released from rest at the top of a frictionless incline. As the block moves through the distance D1, the block gains kinetic energy ΔK1 and gains momentum of magnitude Δp1. As the block moves through the distance D2, the block gains kinetic energy ΔK2 and gains momentum of magnitude Δp2. How do these increases in kinetic energy and momentum compare if D1 = D2? (A) (B) (C) (D)

ΔK2 = ΔK1, Δp2 = Δp1 ΔK2 > ΔK1, Δp2 = Δp1 ΔK2 = ΔK1, Δp2 < Δp1 ΔK2 > ΔK1, Δp2 < Δp1

M6. A light car crashes head-on into a heavier truck. Assume that the only unbalanced forces that each vehicle experiences are due to the other vehicle, and that all motion takes place along a line. Which of the following statements are true as a result of the collision? Select two answers. (A) The car will experience a greater net force than the truck. (B) The car will experience a greater change in momentum than the truck. (C) The car will experience a greater acceleration than the truck. (D) The car will experience a greater change in velocity than the truck.

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Free-Response Questions F1. Students are investigating the principle of conservation of momentum using two unequal-mass carts 1 and 2 that move on frictionless bearings on the same horizontal straight track. The students perform an experimental trial in which cart 1 approaches and collides with cart 2 while motion detectors measure the motion of both carts. The positions of the front of cart 1 and the back of cart 2 are shown as functions of time in the two graphs below.

(a) After taking the above data, but before measuring the masses of the carts, one of the students states that it is clear that cart 2 has more mass than cart 1. How can the student make this determination from the graphs?

The students measure the masses of the carts to be 0.2 kg for cart 1 and 0.7 kg for cart 2. (b) Does the experimental evidence support the law of conservation of momentum? Clearly explain how you analyzed the data to make your determination.

(c) Does the experimental evidence support the assertion that the collision is elastic? Clearly explain how you analyzed the data to make your determination.

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F2. Crossbows are extremely dangerous so do not actually attempt anything that appears in this problem. A crossbow is a device that fires a dart at very high speed. A student wishes to determine that the dart exits the crossbow mechanism. The student cannot simply divide distance by time because no equipment that can accurately measure time is available. The student is only provided with a wooden block, some string, a meterstick, a protractor, and an electronic scale. The student determines that the best course of action is to hang the wooden block from the ceiling by the string and firing the crossbow dart horizontally into the block so that the dart sticks to the block and the block swings upward. (a) In the space below, explain what measurements the student must make in order to calculate the speed of the dart as it exits the crossbow mechanism. Draw a diagram of the experimental setup that shows a labeled geometric quantity that is being measured.

(b) Explain how the student can use the measurements made in part (a) to determine the initial speed of the dart.

(c) The student is not able to vary any aspects of the crossbow, dart, or hanging block. Explain how the student can obtain a more precise measurement of the speed of the dart.

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(d) Let E represent the total mechanical energy of the dart-block-Earth system. i. How does the value of E change as the dart becomes embedded in the block? ___ E increases

___ E decreases

___ E remains the same

Explain your reasoning

ii. How does the value of E change as the dart and block swing upward to their highest point? ___ E increases

___ E decreases

___ E remains the same

Explain your reasoning

(e) Let p represent the total vector momentum of the dart-block system. i. Does p change as the dart becomes embedded in the block? ___ Yes

___ No

Explain your reasoning

ii. Does p change as the dart and block swing upward to their highest point? ___ Yes

___ No

Explain your reasoning

(f) Suppose that, rather than becoming embedded in the block, the dart bounces backward off of the block. Will the block swing to a higher point, lower point, or to the same point as when the dart embedded itself in the block? ___ Higher point

___ Lower point

___ Same point

Explain your reasoning.

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F3. A cart of mass m rolls with initial speed v0 on frictionless bearings. The cart has three slots that fit three identical boxes also of mass m; the slots are equally spaced apart. The cart moves beneath a dispenser that drops one of the boxes directly into each slot. The first box drops into the cart at time T1, the second at time T2, and the third at time T3. (a) On the grid below, draw a graph of the cart’s speed as a function of time. Label the relative values of v0, T1, T2, and T3 on the axes.

(b) In terms of m and v0, write an expression for the final speed of the cart-and-boxes after the final box has dropped into place.

(c) Would the final speed of this cart be different if the slots were spaced closer together? If so, how would the speed be different? In either case, explain your reasoning.

(d) Another student works with a cart also of mass m, but that is much longer and has many slots that can carry boxes. The student suggests that, if enough boxes fall into the cart, the cart will eventually come completely to rest. Is the student’s hypothesis correct? Explain why or why not.

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The students are given a new situation, in which a cart of mass M0 and length D rolls underneath a device dispensing water. The water passes through a hole in the track except during the time that the cart passes underneath the device. The mass of water that empties into the cart every second is R. The cart’s initial speed before passing beneath the water is v0.

(e) A student observing this situation suggests that the final speed of the cart after it leaves the water’s RD downspout is given by the equation v f = v0 − . M0 i. Is it reasonable for the quantity R to appear in the numerator in this equation? Why or why not?

ii. Is it reasonable for the quantity D to appear in the numerator in this equation? Why or why not?

iii. Is it reasonable for the quantity M0 to appear in the denominator in this equation? Why or why not?

iv. Name a specific feature of the equation that prevents the equation from correctly modeling the final speed of the cart.

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F4. A cart with frictionless bearings is released from rest at the top of an incline. The incline is fitted with small bumps that are separated by distances D. The bumps are numbered in sequence as shown in the diagram. The actual track is much longer than shown, with many bumps. A student releases the cart from rest and allows the cart to travel over the first three bumps. The student observes that the cart slows down a little upon traveling over each bump, meaning that the cart loses a little momentum and a little kinetic energy. The student asks this question: “does each bump take from the cart the same amount of momentum or the same amount of kinetic energy?” (a) A second student makes this claim: “The cart gains more momentum in traveling between bumps 1 and 2 than it gains between bumps 2 and 3, but the cart gains the same amount of kinetic energy for both intervals.” Explain why this is the case.

(b) The first student make the following prediction: “If each bump takes away the same small amount of momentum, then the cart will eventually reach a maximum speed regardless of how long it is allowed to roll on the incline. However, if each bump takes away the same small amount of kinetic energy, then the cart will always speed up and never reach a maximum speed.” In a clear, coherent, paragraph-length response which may include equations and/or diagrams, explain why this prediction is correct.

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