# Linear Momentum Lecturer: Professor Stephen T. Thornton

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Linear Momentum Lecturer: Professor Stephen T. Thornton

Reading Quiz A system of particles is known to have a total kinetic energy of zero. What can you say about the total momentum of the system? A) momentum of the system is positive B) momentum of the system is negative C) momentum of the system is zero you cannot say anything about the momentum of the system

Reading Quiz A system of particles is known to have a total kinetic energy of zero. What can you say about the total momentum of the system? A) momentum of the system is positive B) momentum of the system is negative C) momentum of the system is zero D) you cannot say anything about the momentum of the system Since the total kinetic energy is zero, this means that all of the particles are at rest (v = 0). Therefore, since nothing is moving, the total momentum of the system must also be zero.

Conservation of Energy Escape velocity Power Potential energy diagrams
Last Time Conservation of Energy Escape velocity Power Potential energy diagrams

Today Define linear momentum Relationship between K.E. and momentum
More general form of 2nd law Impulse Internal and external forces Collisions

New Concept – Linear Momentum
Linear momentum is simply the product of mass and velocity. Linear momentum is a vector. Sometimes we say just “momentum”. SI unit: kg · m/s

Kinetic energy and linear momentum are intimately related
Kinetic energy and linear momentum are intimately related. Remember this result:

Do demo with bouncing ball and bean bag. One recoils, the other doesn’t.

Change in Momentum

Momentum and Newton’s 2nd Law
A more general form of Newton’s 2nd law:

Let’s see if the equations are consistent.
But when mass is not constant, our new general form should be used.

Impulse What is impulse and why is it useful?
Forces sometimes act between objects over very short times. Examples: Bouncing balls Bat hitting a ball Collisions

The Average Force During a Collision

Definition of Impulse Force can vary considerably over the time of interaction, so let’s consider the average force, :

Impulse is just the change in momentum!

Do egg in a sheet demo.

Conservation of Linear Momentum
What happens when = 0? Then If the net force acting on an object is zero, its momentum is conserved.

Do demos: Rocket bicycle Reaction cars Fire extinguisher rocket cart Water rocket 2-liter bottle rocket

Conservation of Linear Momentum
Law of conservation of linear momentum: When the net external force on a system of objects is zero, the total momentum of the system remains constant. Equivalently, The total momentum of an isolated system remains constant.

Conceptual Quiz An open cart rolls along a frictionless track while it is raining. As it rolls, what happens to the speed of the cart as the rain collects in it? (Assume that the rain falls vertically into the box.) A) speeds up B) maintains constant speed C) slows down D) stops immediately Answer: 3

Conceptual Quiz An open cart rolls along a frictionless track while it is raining. As it rolls, what happens to the speed of the cart as the rain collects in it? (Assume that the rain falls vertically into the box.) A) speeds up B) maintains constant speed C) slows down D) stops immediately Because the rain falls in vertically, it adds no momentum to the box, thus the box’s momentum is conserved. However, because the mass of the box slowly increases with the added rain, its velocity has to decrease. Follow-up: What happens to the cart when it stops raining?

Internal and External Forces
If we have a system of particles, then there can be internal forces (for example, that hold the object together). Internal forces always occur in action-reaction pairs and the sum will be zero.

System of Objects Internal forces have no effect on the net momentum of an object. If the net external force acting on a system is zero, then the net momentum is conserved. Momentum of every particle in system is not conserved, only the net.

Conservation of Momentum
Momentum conservation works for a rocket as long as we consider the rocket and its fuel to be one system and account for the mass loss of the rocket. Figure 9-6. Caption: (a) A rocket, containing fuel, at rest in some reference frame. (b) In the same reference frame, the rocket fires, and gases are expelled at high speed out the rear. The total vector momentum, P = pgas + procket, remains zero.

Rocket Travel. A rocket of total mass 3180 kg is traveling in outer space with a velocity of 115 m/s. To alter its course by 35.0°, its rockets can be fired briefly in a direction perpendicular to its original motion. If the rocket gases are expelled at a speed of 1750 m/s, how much mass must be expelled? Giancoli, 4th ed, Problem 9-7

Conceptual Quiz A) yes B) no
A system of particles is known to have a total momentum of zero. Does it necessarily follow that the total kinetic energy of the system is also zero? A) yes B) no Click to add notes

Conceptual Quiz A system of particles is known to have a total momentum of zero. Does it necessarily follow that the total kinetic energy of the system is also zero? A) yes B) no Momentum is a vector, so the fact that ptot = 0 does not mean that the particles are at rest! They could be moving such that their momenta cancel out when you add up all of the vectors. In that case, because they are moving, the particles would have non-zero KE.

Conceptual Quiz Two boxes, one heavier than the other, are initially at rest on a horizontal frictionless surface. The same constant force F acts on each one for exactly 1 second. Which box has more momentum after the force acts ? A) the heavier one B) the lighter one C) both the same F light heavy Click to add notes

Conceptual Quiz Dp = F Dt , F
Two boxes, one heavier than the other, are initially at rest on a horizontal frictionless surface. The same constant force F acts on each one for exactly 1 second. Which box has more momentum after the force acts ? A) the heavier one B) the lighter one C) both the same av Dt Dp F = , We know: F light heavy so impulse Dp = Fav Dt. In this case F and Dt are the same for both boxes! Both boxes will have the same final momentum.

Conceptual Quiz In the previous question, which box has the larger velocity after the force acts? A) the heavier one B) the lighter one C) both the same Click to add notes

Conceptual Quiz In the previous question, which box has the larger velocity after the force acts? A) the heavier one B) the lighter one C) both the same The force is related to the acceleration by Newton’s Second Law (F = ma). The lighter box therefore has the greater acceleration and will reach a higher speed after the 1-second time interval. Follow-up: Which box has gone a larger distance after the force acts? Follow-up: Which box has gained more KE after the force acts?

Conceptual Quiz A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longer time to bring to rest? A) the bowling ball B) same time for both C) the Ping-Pong ball D) impossible to say p Click to add notes

Conceptual Quiz Dp = F Dt p
A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longer time to bring to rest? A) the bowling ball B) same time for both C) the Ping-Pong ball D) impossible to say av Dt Dp F = We know: p so Dp = Fav Dt Here, F and Dp are the same for both balls! It will take the same amount of time to stop them.

Conceptual Quiz A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, for which is the stopping distance greater? A) the bowling ball B) same distance for both C) the Ping-Pong ball D) impossible to say p Click to add notes

Conceptual Quiz A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, for which is the stopping distance greater? A) the bowling ball B) same distance for both C) the Ping-Pong ball D) impossible to say Use the work-energy theorem: W = DKE. The ball with less mass has the greater speed (why?), and thus the greater KE (why again?). In order to remove that KE, work must be done, where W = Fd. Because the force is the same in both cases, the distance needed to stop the less massive ball must be bigger. p

Conceptual Quiz Amy (150 lbs) and Gwen (50 lbs) are standing on slippery ice and push off each other. If Amy slides at 6 m/s, what speed does Gwen have? A) 2 m/s B) 6 m/s C) 9 m/s D) 12 m/s E) 18 m/s 150 lbs 50 lbs Click to add notes

Conceptual Quiz Amy (150 lbs) and Gwen (50 lbs) are standing on slippery ice and push off each other. If Amy slides at 6 m/s, what speed does Gwen have? A) 2 m/s B) 6 m/s C) 9 m/s D) 12 m/s E) 18 m/s The initial momentum is zero, so the momenta of Amy and Gwen must be equal and opposite. Because p = mv, then if Amy has three times more mass, we see that Gwen must have three times more speed. 150 lbs 50 lbs