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**Conservation of momentum Pg. 33**

This lesson discusses conservation of momentum. In an inquiry activity using a simulation of spring-loaded carts, students are guided to discover of the law of conservation of momentum. This law is shown to be a consequence of Newton’s third law when applied to a closed system. Students learn to apply this law through a set of specific problem-solving steps. Rocket propulsion is discussed as an example of a system in which momentum is conserved even as the mass of an object changes. Ask: What happens if you jump off of a skateboard towards its front? Where does the skateboard go? (The skateboard moves backward.) Ask: Why does this happen?

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**Objectives Physics terms Define the law of conservation of momentum.**

Demonstrate the law of conservation of momentum using an interactive simulation. Apply the law of conservation of momentum in one dimension. law of conservation of momentum Ask students about some of the terms here that they have seen before. Define: momentum; one dimension.

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**Equations Momentum = Momentum before after Momentum:**

Conservation of momentum: Momentum = Momentum before after Equations for this lesson. Momentum equation should be a review from a prior lesson. Conservation of momentum will be a new equation: don’t dwell on it, because it will be introduced later in the lesson.

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**In a closed system, energy is conserved.**

Conservation laws In a closed system, energy is conserved. Ask students to define a system, and what it means for a system to be open or closed. The dotted line depicts the boundary of the system, which will contain the two cars. The system is the same before and after the collision. The word ‘system’ is often used to describe a collection of objects and their interactions.

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**The carts start pinned together**

Conservation laws The carts start pinned together Consider this closed system containing two frictionless carts with opposing springs. The carts must be held together at the beginning or else they will push apart.

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Conservation laws When the pin is released, the carts will fly away from each other. How fast will does each one go? This slide shows the two cars moving away from each other at equal magnitude opposite sign velocities. But what if the masses are not equal?

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**The elastic energy in the two springs . . .**

Energy conservation The elastic energy in the two springs . . . Note that the springs are assumed to be identical, so they both have the same spring constant k and the same compression by distance x. Ask: What kind(s) of energy are present “after” the pin is released? (Answer: kinetic energy.) Ask: Write down the formula for kinetic energy after the pins are released. (Answer on next page.)

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Energy conservation The elastic energy in the two springs equals the kinetic energy of both carts after the release. This slide shows the kinetic energy of the cars after the separation. The slides note that the energies before and after—elastic potential energy and then kinetic energy--are equal. Ask students why the two total energies are equal. (Because of conservation of energy.) Ask students if they can use the information provided above to determine the velocities of the two cars after separation. (Conservation of energy does not provide enough information to determine either of the cars’ velocities.)

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**Energy conservation v1 v2**

This is one equation (conservation of energy) with two unknowns: v1 and v2. Some students may struggle with the algebra concept contained on this slide. Emphasize the basic concept: you cannot solve a problem if you have more unknowns than you have equations. Here there are two unknowns but only one We can’t solve for the final velocities!

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**Not enough information**

v1 v2 Energy conservation does not tell us: 1-whether the carts move at the same or different speeds or 2-if the carts move in opposite directions (though we know they do). Ask: Can you determine the signs of the velocities after separation using conservation of energy? Could both velocities be positive? Could both be negative? Could one be negative and the other positive? Would all these satisfy conservation of energy? (Yes to all.)

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**A second law is needed! Suppose the carts have different masses.**

Are the final speeds still the same? Ask: If the two carts have different masses, then will one move faster after they are released? Ask: How is this related to jumping off of a skateboard? (Your body is high mass, while the skateboard is low mass.)

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A second law is needed! Energy conservation says nothing about how the two velocities compare with each other. Once again, conservation of energy is one equation with two unknowns, so it cannot determine the velocities uniquely.

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**Investigation Student ID login: 742 379 6973**

What determines the final velocities of the carts? Explore this question in Investigation 11A Student ID login: 742 379 6973 -Click on the “investigation” icon -Scroll down & click on Investigation 11A- Conservation of momentum -Select the investigation icon -Complete simulation & answer ?’s Click on the interactive simulation to launch it in a browser. Students should have their assignment sheets to record the data from their investigation.

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Investigation When these ballistic carts are released from rest, the compressed spring causes them to move in opposite directions. Select a mass for each cart. Press [Run] to start. Run the simulation for different combinations of masses for the two carts. Click on the interactive simulation to launch it in a browser (Firefox recommended).

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Investigation Fill in the table based on the different combinations of masses Students should have their assignment sheets to record the data from their investigation.

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**Post Investigation Questions**

Describe the velocities when the masses of the carts are equal. b. Describe the velocities when the red cart has more mass. c. Describe the velocities when the blue cart has more mass. This is the vocabulary term relevant to this lesson.

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**Investigation Evaluate the data in your table.**

What patterns do you see in your data table that is equal and opposite for the two carts after they are released? Wait to move on to the next slides until students have completed their data tables.

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**What patterns do you see?**

Mass, velocity, momentum, and energy data Students should have produced a table with results similar to this one. Ask: Do you see any patterns? Direct students to patterns among equal masses (both 5 kg, both 10 kg, etc.).

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**Equal masses equal speeds**

Mass, velocity, momentum, and energy data Ask: What is the pattern? (Answer: the velocities are equal but opposite in sign. The momenta are also equal but opposite in sign. The kinetic energies are equal.)

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**Twice the mass half the speed**

Mass, velocity, momentum, and energy data Ask: What is the pattern when one cart has more mass than the other? (Answer: the heavy cart moves slower.)

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**Triple the mass 1/3 the speed**

Mass, velocity, momentum, and energy data Similar effect when the masses differ by a factor of three. Ask: How about a pattern for the momentum? (Answer: equal but opposite.) Ask: How about the kinetic energy? (Answer: different values, the more massive cart having less energy.)

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**What principle is operating here?**

Triple the mass 1/3 the speed Mass, velocity, momentum, and energy data What principle is operating here? Notice that the momentum is equal and opposite! Ask: What was the momentum of each cart before they were released? (Answer: both had zero momentum.) Ask: What was their momentum after released? (Answer: equal but opposite.) Ask: What was the total momentum after released? (Answer: zero.) Ask: What do you think is going on with the total momentum before and after they were released?

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**Examining the momentum**

Mass, velocity, momentum, and energy data Ask: What is the total momentum of the system BEFORE the spring is released? What is the total momentum of the system AFTER the spring is released?

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Rocket science How is a rocket launch similar to these spring-loaded carts? What principle launches the rockets? The rocket and the carts obey the same principle! Conservation of momentum Ask: How is a rocket launch similar to these two carts? Hint: For the rocket, what moves in the opposite direction of the rocket? (Answer: the propellant moves in the opposite direction.) As the fuel burns, the mass of the rocket decreases and so the acceleration of the rocket increases. But the momentum of the rocket/fuel system is conserved.

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**Momentum conservation applies equally to . . .**

A universal law Momentum conservation applies equally to . . . rocket engines Extending the analogy to...

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**Momentum conservation applies equally to . . .**

A universal law Momentum conservation applies equally to . . . rocket engines two roller skaters pushing each other apart Extending the analogy to...

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**Momentum conservation applies equally to . . .**

A universal law Momentum conservation applies equally to . . . rocket engines two roller skaters pushing each other apart the interaction of subatomic particles! The interaction of subatomic particles is less familiar, but conservation of momentum can be used to understand subatomic interactions in the same way as macroscopic processes.

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A universal law Momentum is conserved in all interactions between objects, from atoms to . . .

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A universal law Momentum is conserved in all interactions between objects, from atoms to planets! Conservation of momentum is always true, at all scales. It is a fundamental law.

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**The total momentum of a closed system remains constant.**

Conservation of momentum The total momentum of a closed system remains constant. The closed box around the two cars designates that they represent one system and that the conservation of momentum applies to them and no other objects in which they interact. (The springs are included in the system.)

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**Momentum is mass times velocity**

Ask: Is momentum a vector or scalar quantity? (Answer: it is a vector quantity, because it has direction.)

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**The total momentum is zero. Why?**

At the start ... Both initial velocities are zero, so the sum of their momenta is zero. The total momentum is zero. Why?

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**As long as no outside forces act...**

Ask: Are there external forces acting here? Do the springs exert a force when they are released? Are they an internal or external force? (Answer: internal.)

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**The total momentum is conserved.**

As long as no outside forces act... How do the velocities of the two carts differ? (Answer: they differ in direction or sign.) Ask: If each cart is moving at 5 m/s, then how can their momenta add up to zero? (Answer: velocity is a vector quantity, so one of the carts would have to be moving at negative 5 m/s!) Students may ask: but this is also a single equation with two unknowns! Yes, but at least conservation of momentum tells us the correct signs of the velocities, something that energy conservation did not. (How do you calculate the actual velocities? A future lesson describes the connection between force and the rate of change of momentum...) The total momentum is conserved.

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**One cart’s momentum is positive .. .**

Mass, velocity, momentum, and energy data This cart has positive momentum...

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**The other cart’s momentum is negative**

Mass, velocity, momentum, and energy data ...while the second cart has an equal but opposite momentum.

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**The total momentum remains zero.**

If you add equal and opposite momenta, you get zero, which was the momentum we started with. The total momentum remains zero.

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Momentum is a vector Momentum can be positive or negative depending on the direction of the velocity. Ask students to review the definition of a vector; remind them that momentum is a vector because velocity is a vector. (A vector has magnitude and direction.)

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**In 1D, direction is given by the sign of the momentum.**

Negative momentum Positive momentum The sign of the momentum tells you what direction an object moves in for one-dimensional motion. We chose motion in the positive x-direction to correspond to positive momentum, which is usually the convention adopted.

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**The TOTAL force on the system adds to zero.**

Why is the law true? Force Force By Newton’s third law, the carts put equal and opposite forces on each other. The TOTAL force on the system adds to zero.

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**Why is the law true? Since the net force on the system is zero . . .**

Force Force Since the net force on the system is zero . . . the momentum of the system cannot change!

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**Solving momentum conservation problems**

Problem solving steps: Calculate all the known initial momenta for the objects. Calculate all the known final momentum for the objects. Equate the total momentum before to the total momentum after. Solve for the unknown momentum. Allow students time to write down their answer.

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**Apply momentum conservation**

A 6.0 kg package explodes into two pieces, A and B. Piece A, with a mass of 4.0 kg, moves east at 10 m/s. a) What is the mass of piece B? b) What is the direction of piece B? c) What is the speed of piece B? 2.0 kg B A 10 m/s 4.0 kg west Apply conservation of momentum Allow students time to write down their answer.

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**Apply momentum conservation**

A 6.0 kg package explodes into two pieces, A and B. Piece A, with a mass of 4.0 kg, moves east at 10 m/s. What is the speed of piece B? The negative sign means that piece B headed west.

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Assessment Which statement below correctly summarizes the law of conservation of momentum? The momentum of an object always remains constant. The momentum of a closed system always remains constant. Momentum can be stored in objects such as a spring. All of the above. The answer is on the next slide.

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Assessment Which statement below correctly summarizes the law of conservation of momentum? The momentum of an object always remains constant. The momentum of a closed system always remains constant. Momentum can be stored in objects such as a spring. All of the above. Ask the students to explain what is wrong with statements A and C. The momentum of individual objects changes whenever an impulse is applied to them. C. Unlike energy, momentum cannot be stored. Objects have momentum because of their velocity.

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Assessment An astronaut with a mass of 100 kg throws a wrench with a mass of 2.0 kg at a velocity of 5.0 m/s. What is the recoil velocity of the astronaut if both wrench and astronaut are initially at rest? This example is physically similar to the ballistic carts!

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Assessment An astronaut with a mass of 100 kg throws a wrench with a mass of 2.0 kg at a velocity of 5.0 m/s. What is the recoil velocity of the astronaut if both wrench and astronaut are initially at rest? Momentum before = Momentum after throwing wrench throwing wrench The unknown velocity! Before throwing the wrench, the astronaut and wrench were together moving at speed v0. Afterwards, each had its own velocity. Why is this term the unknown momentum quantity? (Answer: the question asks for the astronaut’s final velocity.) Ask: Are any of these quantities equal to zero? (Answer on next slide.)

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Assessment An astronaut with a mass of 100 kg throws a wrench with a mass of 2.0 kg at a velocity of 5.0 m/s. What is the recoil velocity of the astronaut if both wrench and astronaut are initially at rest? Momentum before = Momentum after throwing wrench throwing wrench Initial momentum is zero! The initial velocity was zero, which will make the problem simpler. Since the starting velocity is zero, the initial momentum is zero. Since momentum is conserved, we know that the final momentum must also be zero. The momentum vectors corresponding to the astronaut and the wrench must be equal and opposite.

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Assessment An astronaut with a mass of 100 kg throws a wrench with a mass of 2.0 kg at a velocity of 5.0 m/s. What is the recoil velocity of the astronaut if both wrench and astronaut are initially at rest? Momentum before = Momentum after throwing wrench throwing wrench Subtract the wrench’s momentum from both sides, and then divide both sides by 100 kg to arrive at the answer.

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