LECTURE 23: Momentum-Impulse

WARM UP: With the definition of momentum as mass time velocity, we know that

Momentum is a vector.1.Momentum is a scalar.2.Momentum points in the same direction as velocity.3.The direction of momentum cannot be determined solely by velocity.4.Momentum is a scalar so it does not have a direction.5.

Select LEARNING OBJECTIVES:

TEXTBOOK CHAPTERS:

Giancoli (Physics Principles with Applications 7th) :: 7-1, 7-3•Knight (College Physics : A strategic approach 3rd) :: 9.1, 9.2, 9.3•BoxSand :: Momentum ( Impulse )•

Introduce the concept that objects possess momentum.i.Introduce the concept of impulse.ii.Be able to analyze a force vs time graph, relating area to impulse and change in momentum.iii.Relate kinematic variables such as velocity, acceleration, and displacement of different objects subjected to forces over a given time period.

iv.

Strengthen the ability to determine the relative direction of two vectors if they are related to one another by a scalar.

v.

Strengthen the ability to determine the change in vector quantities.vi.Strengthen the ability to perform vector addition and subtraction.vii.Strengthen the ability to estimate areas under non-linear curvesviii.

We have discussed many different derived quantities thus far, for example, velocity, acceleration, and force. At this time we will introduce a new derived quantity, known as momentum. Momentum is the quantity we get when we scale the velocity of an object by its mass. Mathematically momentum is described as

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Why introduce this new quantity? I mean it is only just the velocity times mass, so how could working in terms of momentum be any more beneficial than working with velocity? Well, consider an object moving at 50 mph. Would you want to get hit by that object? If the object was a ping pong ball, perhaps you wouldn’t mind. But if the object was a brick, then you would definitely not want to get hit by it because we know it will hurt much more than a ping pong ball. So scaling the velocity of an object by its mass provides some more insight into a scenario than just talking about the velocity of an object. Also, if we rewrite Newton's laws in terms of momentum we can greatly simplify the analysis of certain scenarios. But before we explore some of the more useful ways that momentum can help us analyze a scenario, let's first get comfortable with finding momentum for various systems.

EXAMPLE: Two cars are traveling along a straight road as shown below. Consider the system consisting of the two cars. What is the momentum of the system?

PRACTICE: Three cars are traveling along a straight road as shown below. Consider the system consisting of the three cars. What is the momentum of the system?

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PRACTICE: Two hockey pucks are sliding across the ice rink as shown below. Consider the system consisting of the two hockey pucks. The speed of the puck 1 is 10 m/s and the speed of puck 2 is half of speed 1. What is the momentum of the system?

PRACTICE: Three basketball players are shooting hoops and bouncing each shot off the backboard. What is the direction of the change in momentum vector, from before the ball hits the backboard to after, for player one's shot. The dashed lines are the trajectory of the basketballs. (Ignore gravity).

PRACTICE: Assuming the positive x-direction is to the right, the cart's change in momentum is

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PRACTICE: Assuming the positive x-direction is to the right, the cart's change in momentum is

PRACTICE: Three basketball players are shooting hoops and bouncing each shot off the backboard. Sketch the change in momentum vector for player two assuming the speed is the same before and after hitting the backboard. (Ignore gravity)

PRACTICE: Three basketball players are shooting hoops and bouncing each shot off the backboard. Sketch the change in momentum vector for player three assuming the speed is reduced during the bounce off the backboard. (Ignore gravity)

A change in momentum is given a name called, impulse, and is represented by the letter J. But remember

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A change in momentum is given a name called, impulse, and is represented by the letter J. But remember momentum is a vector, and a change in momentum is also a vector, thus impulse is a vector too. Mathematically we write this as…

Now let's revisit Newton's 2nd law of motion.

Recall the definition of average acceleration is the change in velocity divided by the change in time. Using this definition we can recast Newton's 2nd law as…

Making note that we are now working with average quantities because of the finite nature of our average acceleration definition. As a side note, if we don't assume that the mass of the object we are analyzing with the second law is constant, we could include it with the change in velocity. Now Newton's 2nd law takes on a form that relates net external forces acting as a cause to an changes in momentum being the effect.

Alternatively, we can rearrange the above mathematical equation as follows…

This form is fascinating! It highlights the notion that the net external force acting on an object for some delta t time can cause changes in momentum. This shouldn't be too strange of a statement; if you push a block initially at rest for a few seconds, the mass of the block stayed the same but the velocity changed so therefore the momentum also changed.

Now recall that we had just defined a new word for changes in momentum, impulse. If we make all of these connections between changes in momentum, forces over time, and impulse, we get what is known as the "impulse-momentum theorem".

What this theorem tells us is that the change in momentum (impulse) of a system is equal to the average net external force acting on the system multiplied by the change in time that the external force was applied.

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PRACTICE: The diagram depicts two pucks on a frictionless table. Puck 2 is four times as massive as puck 1. Starting from rest, the pucks are pushed across the table by two equal forces. The forces act on both of them for 6.0 seconds.

Rank the final momentum of the two pucks.

PRACTICE: The diagram depicts two pucks on a table with friction. Puck 2 is four times as massive as puck 1. Starting from rest, the pucks are pushed across the table by two equal forces. The forces act on both of them for 6.0 seconds.

Rank the final momentum of the two puck

PRACTICE: The diagram depicts two pucks on a frictionless table. Puck 2 is four times as massive as

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PRACTICE: The diagram depicts two pucks on a frictionless table. Puck 2 is four times as massive as puck 1. Starting from rest, the pucks are pushed across the table by two equal forces. The forces act on both of them all the way to the finish line.

Which puck will have the greater momentum upon reaching the finish line?

PRACTICE: A 50 g rubber ball and a 50 g clay ball are thrown at a wall with equal speeds. The rubber ball bounces, the clay ball sticks. Which ball exerts a larger impulse on the wall?

They exert equal impulses because they have equal momenta.(1)The clay ball exerts a larger impulse because it sticks.(2)Neither exerts an impulse because the wall doesn't move.(3)The rubber ball exerts a larger impulse because it bounces.(4)

Graphical analysis of impulse-momentum theorem

Take a close look at impulse, it is equal to a net force external on the system times a change in time. Thus, if we create a graph of net force versus time, the area would be impulse. This is seen in the graph below…

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The above graph could represent the x-component of force a ball feels as it hits and bounces off of a wall. The green curve would be a more realistic shape, which we could confirm experimentally, however, we could approximate this time dependent "green" net force as a constant "red" net force that we call the force. The area under both of these respective graphs gives the impulse, or change in momentum in the x-component. Both areas in this case are equal, so the change in x-component of momentum is the same.

PRACTICE: What is the impulse delivered to a 5 kg object that experiences the forces below?

PRACTICE: The graph below shows the average net force acting horizontally on a 0.16 kg billiard ball vs time as it hits a wall. What is the final velocity of the ball assuming the initial velocity was zero.?

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PRACTICE: If the billiard ball was made out of a softer material and went through the same change in momentum after being hit, what would the new force vs time graph look like?

Questions for discussion

Consider a car going around a circle at a constant speed. Is the moment of the car also constant?1)Will a change in momentum always have the same direction as the change in velocity for an object?

2)

If the external forces acting on your object are not constant in time, can you use to

find the change in momentum? 3)

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