Chapter 7 Linear Momentum and Impulse Notes © 2014 Pearson Education, Inc.

Momentum & Force Momentum is a vector symbolized by the symbol p, and is defined as The rate of change of momentum (known as impulse) is equal to the net force: Newton’s 2 nd Law: The rate of change of momentum of an object is equal to the net force applied to it © 2014 Pearson Education, Inc.

Law of Conservation of Momentum The total momentum of an isolated system of objects remains constant. An isolated system has no EXTERNAL forces acting on it. Example: Tennis ball and racket Isolated system: only in the brief instants before and after the collision External forces (gravity, air resistance) act on this system over a longer time scale

Other Examples 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. Momentum of fuel out the back balances the forward momentum of the rocket. © 2014 Pearson Education, Inc.

Other Examples Would momentum be conserved or not? If frictionless track – yes. In real world – no. If air, track, earth, etc are all part of the system – yes.

Other Examples Would momentum be conserved or not? Depends on the definition of the system: If system = rock and person then no because gravity accelerates it (external force) If system = rock, earth, and person then yes because no external forces and the earth/person accelerates up and canceling the downward acceleration of the rock

Solving Conservation of Momentum © 2014 Pearson Education, Inc.

Collisions and Forces During a collision, the system’s internal forces are usually much greater than external forces, so we can ignore external forces for the brief interval of the collision Objects are deformed due to the large forces involved © 2014 Pearson Education, Inc.

When the collision occurs, the force jumps from zero to a very large force in a short time © 2014 Pearson Education, Inc. Though the force is not constant, we can approximate the total force by using average force over the entire time interval

During a collision, the force is equal to the change in momentum (impulse) divided by time Impulse-Momentum Theorem: © 2014 Pearson Education, Inc. Impulse Impulse is the green shaded area

What is the point of these safety items in light of impulse? Impulse in the Real World Change in momentum is constant (from whatever mv 0 to zero), so increase time it takes, results in decreased force

The impulse tells us that we can get the same change in momentum with a large force acting for a short time, or a small force acting for a longer time. This is why you should bend your knees when you land; why you wear boxing gloves; why landing on a pillow hurts less than landing on concrete, why a karate chop has to be FAST, and why a slap stings

Practice Problems 7-1. For a top player, the tennis ball must leave the racket on a serve with a speed of 55m/s (about 120mph). If the ball has a mass of 0.060kg and is in contact with the racket for about 4ms (4 x s), a) estimate the average force on the ball. b) Would this force be large enough to lift a 60kg person? C) compare this to the force of gravity on the ball to justify why it can be ignored at the moment of the collision.

Practice Problems 7-2. Water leaves the hose at a rate of 1.5kg/s with a speed of 20m/s and is aimed at the side of a car, which stops it. (Ignore splashing back) What is the force exerted by the water on the car?

Practice Problems 7-3. A 10,000kg railroad car, A, traveling at a speed of 24.0m/s strikes an identical car, B, at rest. If the cars lock together as a result of the collision, what is their common speed afterward?

Practice Problems 7-4. A) an empty sled is sliding on frictionless ice when Susan drops vertically from a tree down onto the sled. When she lands, does the sled speed up, slow down, or keep the same speed? B)Later, Susan falls sideways off the sled. When she drops off, does the sled speed up, slow down, or keep the same speed?

Practice Problems 7-5. Calculate the recoil velocity of a 5.0kg rifle that shoots a 0.020kg bullet at a speed of 620m/s.

Practice Problems 7-6. Estimate the impulse and the average force delivered by a karate blow that breaks a board. Assume the hand is 1kg and moves at roughly 10m/s when it hits the board.

Conservation of Energy in Collisions Notes © 2014 Pearson Education, Inc.

Momentum is conserved in all collisions. Collisions in which kinetic energy and internal energy are conserved are called elastic collisions, and those in which KE and internal E are not conserved are called inelastic. © 2014 Pearson Education, Inc.

Elastic Collisions When two objects collide, return to their original shapes and move separately after the collision Momentum is still conserved Kinetic energy is conserved in elastic collisions, internal E of objects constant Elastic collisions don’t truly exist in the real world – i.e. if there is sound, then KE has been transformed

Elastic Collisions Equations m A v A,0 + m B v B,0 = m A v A + m B v B K A,0 + K B,0 = K A + K B,0 In a perfectly elastic collision between balls of equal masses, the balls exchange velocities

Inelastic Collisions A collision in which two objects stick together and move together after colliding, typically generate thermal energy which changes internal energy of the objects Momentum is still conserved

Kinetic energy is not conserved in inelastic collisions – Some energy is transformed into sound, heat, potential, etc. – Kinetic energy could also be gained such as in an explosion (inelastic collision in reverse) when potential nuclear or chemical energy is transformed

Perfectly Inelastic Collision Formula m A v A,0 + m B v B,0 = (m A + m B )v

Collisions in 2 or 3 Dimensions Conservation of energy and momentum can also be used to analyze collisions in 2D or 3D, but angles are necessary in addition to masses and initial velocities © 2014 Pearson Education, Inc.

Center of Mass Until now, we have assumed all objects behave like a particle that undergoes only translational motion (as in figure a) However, real objects also rotate and move in other ways that are not just translational – this is called general motion But there is one point in the object that despite general motion, that one point only has translational motion like a particle (as in figure b) Center of Mass (CM) – the point in an object that behaves like a particle © 2014 Pearson Education, Inc.

The center of mass continues to move according to the net force The general motion of an object or system is the sum of the translational motion of the CM, plus rotational, vibrational, or other forms of motion about the CM. © 2014 Pearson Education, Inc.

The velocity of the CM of a system cannot be changed by an interaction in the system. (Velocity of CM in these situations is always zero) © 2014 Pearson Education, Inc. Which is a correct final location?

When two objects collide, the velocity of their center of mass will not change. © 2014 Pearson Education, Inc. If m 1 =m 2 and v 1 =v 2, then momentum and velocity of CM are zero before and after the collision

The center of mass can be found experimentally by suspending an object from different points. The CM need not be within the actual object (i.e. a doughnut’s CM is in the center of the hole) © 2014 Pearson Education, Inc. High jumpers have developed a technique where their CM actually passes under the bar as they go over it. This allows them to clear higher bars.

Practice Problems 7-7 (modified to a concept question). A) Ball A and B have equal 2.0kg masses. Ball A has an initial velocity of 5.0m/s and Ball B has an initial velocity of -10.0m/s. What are the final velocities of the two balls after they collide. B) Now Ball B is initially at rest. What is the velocity of each ball after they collide?

Practice Problems A 10,000kg railroad car, A, traveling at a speed of 24.0m/s collides with an identical car, B, which is at rest. The cars lock together and are moving at 12.0m/s after the collision. How much energy was transformed into thermal and other types of energy during this collision?

Practice Problems 7-12 (modified). Three guys of roughly the same mass are on a raft. A) Approximately where is the CM of the system? B) What would happen if the middle guy moved backward to 2.0m?

Practice Problems 7-14 (modified). A rocket is shot into the air as shown. At the moment the rocket reaches its highest point, a horizontal distance 10.0m from its starting point, a prearranged explosion separates the equal masses. Part I is stopped in midair by the explosion and falls vertically to Earth. Where does part II land compared to the starting point?