Chapter 9 Linear Momentum Demos Finish up pendulum demo by deriving restoring force. B. Find cm demo for Virginia. C. Air track with colliding cars with springs in between them. Two cars of the same mass with spring between them shows that cm does not move. Demo comparing an inelastic and elastic collision

Assignment 3 Chp 6-2, 6-57, 6-62 9-13, 9-34, 9-54, 9-62, 9-83 Due Monday 9:00 AM

Topics Momentum and Its Relation to Force Conservation of Momentum Collisions Conservation of Energy and Momentum in Collisions Elastic & Inelastic Collisions in One Dimension Elastic & Inelastic Collisions in Two Dimensions

Center of Mass (CM) Center of Mass and Translational Motion Systems of Variable Mass; Rocket Propulsion

Linear Momentum What is momentum and why is it important? Momentum p is the product of mass and velocity for a particle or system of particles. The product of m and v is conserved in collisions and that is why it is important. It is also a vector which means each component of momentum is conserved. It has units of kg m/s or N-s.

Linear Momentum form of Newton’s 2nd Law Now note we have also a force when the mass changes

Law of Conservation of Linear Momentum http://www.colorado.edu/physics/2000/bec/

Therefore each component of the momentum Px, Py, Pz is also constant. Gives three equations: Px= constant Py = constant Pz = constant If one component of the net force is not 0, then that component of momentum is not a constant. For example, consider the motion of a horizontally fired projectile. The y component of P changes while the horizontal component is fixed after the bullet is fired.

Conservation of Momentum Conservation of momentum holds during collisions. A collision takes a short enough time that we can ignore external forces. Since the internal forces are equal and opposite, the total momentum is constant. Figure 9-4. Caption: Collision of two objects. Their momenta before collision are pA and pB, and after collision are pA’ and pB’. At any moment during the collision each exerts a force on the other of equal magnitude but opposite direction.

Types of Collision Two colliding cars. Rubber ball bouncing off the floor Two balls colliding on a pool table. Tennis racquet striking a ball A bullet striking a target. High speed electron striking a proton

Conservation of Momentum Example Railroad cars collide and stick: momentum conserved ? Yes. A 10,000-kg railroad car, A, traveling at a speed of 24.0 m/s strikes an identical car, B, at rest. If the cars lock together as a result of the collision, what is their common speed immediately after the collision? Figure 9-5. Solution: Momentum is conserved; after the collision the cars have the same momentum. Therefore their common speed is 12.0 m/s.

Explosions Opposite of a collision Chemical or nuclear explosion Exploding bomb Firing a bullet Critical Mass

Conservation of Momentum Example: Rifle recoil. Calculate the recoil velocity of a 5.0-kg rifle that shoots a 0.020-kg bullet at a speed of 620 m/s. Figure 9-7. Solution: Use conservation of momentum. The recoil velocity of the gun is -2.5 m/s.

What happens during a collision on a short time scale? F(t) What happens during a collision on a short time scale? Consider one object the projectile and the other the target.

J is called the impulse Change in momentum of the ball. J is a vector Also you can “rectangularize” the graph

Shape of two objects while colliding with each other. Obeys Newtons third Law

Racquet and Tennis Ball Collision Figure 9.8 Racquet and Tennis Ball Collision

Example: Andy Rodick has been clocked at serving a tennis ball up to 149 mph(70 m/s). The time that the ball is in contact with the racquet is about 4 ms. The mass of a tennis ball is about 300 grams. What is the average force exerted by the racquet on the ball?

b) What is the acceleration of the ball? c) What distance does the racquet go through while the ball is still in contact?

One Dimension Head-on Collision Total momentum before = Total momentum after True for any collision

One Dimension Elastic Collision Kinetic energy before=Kinetic energy after True only for elastic collision

This is important to know because you can solve problems using two equations linear in velocity. This replaces Kinetic Energy equation. Giancoli Notation

Problem 9-34 Giancoli A 0.060 kg ball moving with speed 4.50 m/s has a head–on collision with a 0.090 kg ball initially moving in the same direction with speed 3.00 m/s. Determine the final speed and direction of each ball assuming an elastic collision.

Some simple results for a head-on collision with v2i=0

You want to understand 3 cases m1=m2 m2>>m1 m2<<m1

Balls bouncing off massive floors, we have m2 >>m1

Colliding pool balls The executive toy m2 = m1 Why don’t both balls go to the right each sharing the momentum and energy?

Bowling pin example m2<<m1

Types of Collisions Elastic Collisions: Kinetic energy and momentum are conserved Inelastic Collision: Only P is conserved. Kinetic energy is not conserved Completely inelastic collision. Masses stick together P is conserved

Completely Inelastic Collision Now look at kinetic energy Note Ki not equal to Kf

Completely Inelastic For equal masses Kf = 1/2 Ki, we lost 50% of Ki Where did it go? It went into energy of binding the objects together, such as internal energy, rearrangement of the atoms, thermal, deformation sound, etc.

Collisions in Two Dimensions with particle 2 at rest

2 Dimensional Elastic Collisions with particle 2 at rest Write down conservation of momentum in x and y directions separately. Two separate equations because momentum is a vector. Write down conservation of kinetic energy equation. 3 equations in 2 dimensions

Problem 9-88

Problem 90-88 A bowling ball travelling at 13.0 m/s has 5 times the mass of a pin and the pin goes off at 75 degrees. Find the speed of the pin, speed of ball, and the angle of the ball.

Improving your Billiards

How do you know what angle the billiard ball deflects in an elastic collision? Head on Arbitrary Angle 90 Degrees Line of Action

Pool shot pocket 1 2 Assuming no spin Assuming elastic collision

Where do you aim the Bank shot to make the pocket? Assuming no english on the balls Assuming elastic collision and no bank deformation d

How high does a ball bounce up after an almost elastic collision between floor and bouncing ball? Initial hf Drop a ball from height h. Its initial energy is all potential, equal to mgh. Just before it collides with the floor, it energy is all kinetic. Just after the bounce, if no friction forces have acted, its energy is the same as before, and is all kinetic, therefore its speed must be the same. As it rises it converts its kinetic energy to potential energy, doing the reverse of what it did during the fall. The ball rises to the same height it had initially. This is called an elastic bounce, or elastic collision. In reality no ball is perfectly elastic. Some energy is lost to friction during the collision with the floor when the ball, and also the floor are distorted by the force between them needed to accelerate the ball. SHOW balls of varying elasticity bouncing. SHOW double ball bounce. Does this violate energy conservation? What is happening here? -vi Before bounce vf After bounce

Almost elastic collision Measuring velocities and heights of balls bouncing from a infinitely massive hard floor Type of Ball Coefficient of Restitution (C.O.R.) Rebound Energy/ Collision Energy (R.E./C.E.) Superball 0.90 0.81 Racquet ball 0.85 0.72 Golf ball 0.82 0.67 Tennis ball 0.75 0.56 Steel ball bearing 0.65 0.42 Baseball 0.55 0.30 Foam rubber ball 0.09 Unhappy ball 0.10 0.01 Beanbag 0.05 0.002 Almost elastic collision Almost inelastic collision

Two moving colliding objects: HRW Problem 45 ed 6 Show demo comparing the height of a ball bouncing off the floor compared to one bouncing off another ball. Explain. m1 m2 DEMO SHOWING SUPERBALL FALLING ON TOP OF BASKETBALL. WHY DOES BALL JUMP SO MUCH HIGHER? DEMO SHOWING BASEBALL FALLING ON TOP OF BASKETBALL. WHAT MASS IS NEEDED SO THAT BOTH BALLS STOP DEAD? Just after the little ball bounced off the big ball Just after the big ball bounced off the floor Just before each hits the floor v1f v m1 -v -v v m2 How high does superball go compared to dropping it off the floor?

v1f -v -v v v m1 m2 Just after the big ball bounced off the floor DEMO SHOWING SUPERBALL FALLING ON TOP OF BASKETBALL. WHY DOES BALL JUMP SO MUCH HIGHER? DEMO SHOWING BASEBALL FALLING ON TOP OF BASKETBALL. WHAT MASS IS NEEDED SO THAT BOTH BALLS STOP DEAD? Just after the big ball bounced off the floor Just after the little ball bounced off the big ball Just before each hits the floor v1f v m1 -v -v v m2

-v v For m2 =3m1 v1f = 8/4V =2V superball has twice as much speed. 4 times higher

-v v For maximum height consider m2 >>m1 How high does it go? V1f=3V V2f = -V How high does it go? 9 times higher How about 3 balls?

Center of Mass

Center of mass (special point in a body) Why is it important? For any rigid body the motion of the body is given by the motion of the cm and the motion of the body around the cm. The motion of the cm is as though all of the mass were concentrated there and all external forces were applied there. Hence, the motion is parabolic like a point projectile. How do you show projectile motion is parabolic?

What happens to the ballet dancers head when she raises her arms at the peak of her jump? Note location of cm relative To her waist. Her waist is lowered at the peak of the jump. How can that happen?

Center of Mass As an example find the center of mass of the following system analytically. Note that equilibrium is achieved when the lever arm time the mass on the left equals the similar quantity on the right. If you multiply by g on both sides you have the force times the lever arm, which is called torque. d . m1 m2 x xcm

Center of Mass y x1 m1 m2 xcm x2

Problem 2 dimensions Find xcm and ycm

How do you find the center of mass of an arbitrary shape? Show demo

How do you find Center of Mass for a system of particles analyticaly

Newton’s Second Law for a System of particles: Fnet= Macm take d/dt on both sides take d/dt again Identify ma as the force on each particle

Linear Momentum form of Newton’s 2nd Law for a system of particles Important - a vector quantity that is conserved in interactions. Now take derivative d/dt of

Linear Momentum form of Newton’s 2nd Law for a system of particles Total momentum is conserved for a system of particles if

For a collision the velocity of cm is also constant when no external forces are present The velocity of the cm is total momentum /total mass. In general Consider the total inelastic collision In the initial state Vcm

Did the velocity of the center of mass stay constant for the inelastic collision? For equal mass objects After the inelastic collision what is Vcm. Since the particles are stuck together it must be the velocity of the stuck particles which is v1i/2. Hence, they agree.

Problem 9-32 ed 6 HRW A man of mass w is at rest on a flat car of mass W moving to the right with speed v0. The man starts running to the left with speed vrel relative to the flatcar. What is the change in the velocity of the flatcar Δv = v-v0? Note W and w are mass

Initial momentum = Final momentum vrel v0 vg v Initial momentum = Final momentum Solve for v-v0

Problem 9-32 ed 6 vrel v0 What is the change in the velocity of the car Δv =v-v0? Initial momentum = Final momentum Basis for the rocket equation

Problem The balloon and man is stationary. Vcm=0 The man starts walking up the ladder with speed vrel relative to the balloon. In what direction does the balloon move? What speed does it move at? What is the speed when the man stops? How far down did the balloon move? vrel. m

Problem Balloon moves downward to keep cm fixed b) What speed does the balloon move at? c) The balloon stops because his internal motion stops. vB=0 d)Balloon moves= M uMg vrel m

Systems of variable Mass Derivation of Rocket Equation

Derivation of rocket equation Both dM/dt and vrel are negative. This is like the force pushing the rocket

vrel v

When external forces are present See Giancoli here

Example 9-20 A rocket has a mass of 21000 kg of which 15000 is fuel. The fuel is spewed out at a rate of 190 kg/s with a speed of 2800 m/s. If the rocket is launched vertically upward, find The thrust Net force at blast off Velocity as a function of time Final velocity at burn out

Problem grain B=540 kg/min v =3.20 m/s What external force is needed to keep the railroad car moving at constant speed? The grain that is being added to the car is changing the mass of the car at the rate of 540 kg/min. There is friction between The grain and cart.

Problem grain B=+540 kg/min v =3.20 m/s x vrel is the relative velocity of the grain relative to the cart in the x direction. vrel = 0 - v dm/dt is the rate at which mass is changing

A rope of length L lies in a straight line on a frictionless table, except for a very small piece at one end which hangs down through a hole in the table. This piece is released, and the rope slides down through the hole. What is the speed of the rope at the instant it loses contact with the table? Use energy conservation and concept of cm L L/2

2nd Solution

(b) A rope of length L lies in a heap on a table, except for a very small piece at one end which hangs down through a hole in the table. This piece is released, and the rope unravels and slides down through the hole. What is the speed of the rope at the instant it loses contact with the table? (Assume that the rope is greased, so that it has no friction with itself.) x L