CHAPTER 7 Impulse and Momentum

Objective Define and calculate momentum. Describe changes in momentum in terms of force and time. Source: Wikimedia Commons

Linear Momentum

When we think of a massive object moving at a high velocity, we often say that the object has a large momentum. A less massive object with the same velocity has a smaller momentum. On the other hand, a small object moving with a very high velocity has a large momentum.

Linear Momentum The faster an object moves, the more momentum the object has and the more difficult it is for it to come to a stop. The more massive an object is, the more force it will exert on another object (during a collision) because of its momentum.

Examples: Momentum A 21 kg child is riding a 5.9 kg bike with a velocity of 4.5 m/s to the northwest. What is the momentum of the child alone? What is the momentum of the bike alone? What is the momentum of the bike and child together? What velocity would a child with a mass of 42 kg need in order to have the same momentum?

Examples: Momentum (a)

Examples: Momentum (b)

Examples: Momentum (c)

Examples: Momentum (d)

Impulse

7.1 The Impulse-Momentum Theorem

final momentuminitial momentum IMPULSE-MOMENTUM THEOREM When a net force acts on an object, the impulse of this force is equal to the change in the momentum of the object impulse

7.1 The Impulse-Momentum Theorem Example 2 A Rain Storm Rain comes down with a velocity of -15 m/s and hits the roof of a car. The mass of rain per second that strikes the roof of the car is kg/s. Assuming that rain comes to rest upon striking the car, find the average force exerted by the rain on the roof.

7.1 The Impulse-Momentum Theorem Neglecting the weight of the raindrops, the net force on a raindrop is simply the force on the raindrop due to the roof.

7.1 The Impulse-Momentum Theorem Conceptual Example 3 Hailstones Versus Raindrops Instead of rain, suppose hail is falling. Unlike rain, hail usually bounces off the roof of the car. If hail fell instead of rain, would the force be smaller than, equal to, or greater than that calculated in Example 2?

Example: A force of 2 N applied over a period of 4 s delivers an impulse of 8 Ns 2 N The impulse is a vector. It has the same direction as the applied force. Impulse

Impulse – momentum theorem Impulse-momentum theorem state that: A change in momentum takes force and time Force x (change in time) = change in momentum Change in momentum is proportional to force Change in momentum is proportional to time interval

Impulse – momentum theorem uses Used to determine stopping time and safe following distances for cars and trucks. Used to design safety equipment that reduces the forces exerted on a human body during collisions

On a trampoline, jumpers are protected from injury because the rubber reduces the force of the collision by allowing it to take place over a longer period of time. Increasing time of impact

Egg Impulse Demo

Impulse Graphically, the impulse is the area under a force vs. time graph. F (N) t (s) J

Impulse Even if the force varies, the same principle should hold: impulse is the area under the force vs. time curve. t (s) F (N) J

Example Problem 1: Force and momentum A 1400kg car moving westward with a velocity of 15m/s collides with a utility pole and is brought to rest in 0.30s. Find the magnitude of force exerted on the car during the collision?

Example Problem 1: Force and momentum

Example Problem 2: Stopping time and distance How long does it take to stop a car with a mass of 2400 kg travelling at 20 m/s, if the stopping force is 8000N? How far does the car travel in that time?

Example Problem 2: Stopping time and distance

DUE IN CLASS TOMORROW Assignment

CONSERVATION OF MOMENTUM SECTION 2

A child playing in her garage after a snowstorm, gives a wagon a shove, so that the wagon rolls out the garage door with a speed of 4.0 m/s. As the wagon passes under the door, a load of snow with a mass of 3.0 kg falls off the roof of the garage straight down into the wagon. What is the speed of the wagon after the snow falls into it? The wagon has a mass of 10.0 kg. The wagon can be considered to roll without friction. Conservation of Momentum Practice V i = 4.0

Conservation of Momentum Practice Consider the wagon and snow to be a single system. Since no horizontal external forces act on this system, the horizontal momentum is conserved. FNFN F G-SNOW F G-WAGON

Before After

Conservation of Momentum In its most general form, the law of conservation of momentum can be stated as follows: The total momentum of all objects interacting with one another remains constant regardless of the forces between the objects Regardless of the nature of the forces between the objects, Momentum is conserved in collisions

Conservation of Momentum

The diagram below shows the velocity and momentum of each soccer ball both before and after the collision. The momentum of each ball changes due to the collision, but the total momentum of the two balls together remains constant

Conservation of Momentum In other words, the momentum of ball A plus the momentum of ball B before the collision is equal to the momentum of ball A plus the momentum of ball B after the collision. when you use conservation of momentum to solve a problem or to investigate a situation, it is important to include all objects that are involved in the interaction.

Conservation of Momentum

Practice problem 1 A 44 kg student on in-line skates is playing with a 22 kg exercise ball. The student, starting at rest, throws the ball, causing the student to glide back at 3.5 m/s. How fast was the ball moving away from the student?

Example Problem 1: conservation of momentum

Practice Problem 2 An 85.0 kg fisherman jumps from a dock into a 135 kg rowboat at rest on the west side of the dock. If the velocity of the fisherman is 4.30 m/s to the west as he leaves the dock, what is the final velocity of the fisherman and the boat?

Example Problem 2: conservation of momentum

7.2 The Principle of Conservation of Linear Momentum Example 6 Ice Skaters Starting from rest, two skaters push off against each other on ice where friction is negligible. One is a 54-kg woman and one is a 88-kg man. The woman moves away with a speed of +2.5 m/s. Find the recoil velocity of the man.

7.2 The Principle of Conservation of Linear Momentum

CENTER OF MASS The motion of the

Objects and Systems An object is something that can be treated as having no internal structure. Some elementary particles, like electrons, have no internal structure, as far as we know. Other things have internal structure, but it is not relevant to the question at hand. A point object has no shape. It occupies a specific point. A system is an object or a collection of objects.

CM Source: Zhatt, “Orbit3.gif”, Wikimedia Commons

The center of mass is the “weighted” average position of the particles in the system. The center of mass is in the geometric center of a uniformly dense object. Center of Mass CM

The center of mass of a symmetrical shape is located on the axis or plane of symmetry. The CM of a sphere is the center. The center of mass of a more complex arrangements of shapes can be found by finding the CM of each partial shape, and then finding the CM of the whole system. Center of mass of a symmetrical shape with uniform density 2 kg

Consider when forces are applied to objects that are part of a system. Lighter objects accelerate more for the same force, but their acceleration has less effect on the acceleration of the center of mass. It can be shown that Newton’s 2 nd Law applies to the motion of the center of mass of a system Forces Acting On a System

According to Newton’s Third Law, for every internal force, there will be a corresponding equal force in the opposite direction. Thus, all the internal forces cancel out, and only external forces determine the motion of the center of mass. Forces Acting On a System

Example 1 A small spaceship is at rest in outer space. The astronaut inside stands up and runs from the back of the spaceship to the front at a speed of 3 m/s, where he sits back down. Which of the following best describes the motion of the spaceship during and after the motion? A) The ship did not move either during or after the motion. B) The ship moved backwards during and after the motion. C) The ship moved backwards during the motion, but remained motionless afterward. D) The spaceship moved backwards during the motion, and forwards after.

A blue Angry Bird is fired into the air with an initial velocity of 20 m/s at an angle of 30 degrees. At some point in its flight, it splits into three equal sized Angry Birds. The Angry Bird that lands closest to the launch site lands at 24.6 meters away from the launcher. Where do the other two birds land? Example 2: © Rovio

Motion of the center of mass The center of mass (c.m) of a system is the point where the system can be balanced

The center of mass of a system of objects obeys Newton’s 2 nd Law Consider an artillery shell fired from a cannon. The center of mass of the shell follows a parabolic trajectory Once the shell explodes, the cluster of fragments form a cloud about the center of mass which will follow the original trajectory of the shell. Motion of the center of mass

The total momentum remains constant since no external forces act on the shell. Ex. Astronaut on a spacewalk throws a rope around an asteroid, then pulls the asteroid towards him where will the asteroid and astronaut collide? At the center of mass There are no external forces so the c.m will remain at rest until they collide Motion of the center of mass

Assignment Worksheet

7.2 The Principle of Conservation of Linear Momentum Applying the Principle of Conservation of Linear Momentum 1. Decide which objects are included in the system. 2. Relative to the system, identify the internal and external forces. 3. Verify that the system is isolated. 4. Set the final momentum of the system equal to its initial momentum. Remember that momentum is a vector.

On your conservation of momentum WS -1 You cannot yet answer question 11, we have not discussed center of mass Any questions on the h/w? If not, turn it in

ELASTIC AND INELASTIC COLLISION SECTION 3 Notes have been modified, so you will need some extra notebook paper for the modified sections

Collisions In a collision, it is often difficult to know the details of the forces and motions of the interaction. We can still use conservation laws to draw conclusions about the motions of objects before and after collisions.

Kinetic Energy The energy of a moving object is called kinetic energy. KE = ½ m v 2 New slide

Restoration of Kinetic Energy In some collisions, the total kinetic energy after the collision equals the kinetic energy before the collision. This is sometimes called conservation of kinetic energy. A more accurate phrase is restoration of kinetic energy. New slide

Collisions There are 2 major types of collisions: Perfectly inelastic collisions – when 2 objects collide and move together as one mass (coupled). Elastic collisions – 2 objects collide and return to their original shapes with no change in total kinetic energy. After the collisions, the objects move off separately.

Perfectly Inelastic Collisions

Elastic Collisions

a.A moving ball strikes a ball at rest. b.Two moving balls collide head-on, and move on in opposite directions, momentum is conserved c.Two balls moving in the same direction collide.

7.3 Collisions in One Dimension The total linear momentum is conserved when two objects collide, provided they constitute an isolated system. Elastic collision -- One in which the total kinetic energy of the system after the collision is equal to the total kinetic energy before the collision. Inelastic collision -- One in which the total kinetic energy of the system after the collision is not equal to the total kinetic energy before the collision; if the objects stick together after colliding, the collision is said to be completely inelastic.

Two types of collisions Elastic collisions The objects bounce during the collision so that kinetic energy is restored. The objects separate after the collision. Momentum is conserved. Inelastic collisions The objects deform during the collision so that kinetic energy is transformed to sound or deformation. The objects may stay together or separate after the collision. If they move together, that is a perfectly inelastic collision. Momentum is conserved. New slide

Practice Problem – Inelastic collision

Practice Problem – Elastic collision A kg marble moving to the right at m/s makes an elastic head-on collision with a kg shooter marble moving to the left at m/s. After the collision, the smaller marble moves to the left at m/s. Assume that both marbles are moving on a frictionless surface. What is the velocity of the kg marble after the collision? Show that the change in KE is 0J

ASSIGNMENT Worksheet