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Chapter 7 Momentum and Collisions

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Momentum Newton’s Laws give a description of forces ○ There is a force acting or their isn’t ○ But what about in between Constant velocity Momentum is a quantity defined by the velocity of a mass ○ Velocity changes -> Force Impulse Momentum describes collisions ○ Magnitude gives an indication of how hard it is to stop ○

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Momentum FFrom Newton’s laws: force must be present to change an object’s velocity (speed and/or direction) WWish to consider effects of collisions and corresponding change in velocity MMethod to describe is to use concept of linear momentum scalar vector Linear momentum = product of mass velocity Golf ball initially at rest, so some of the KE of club transferred to provide motion of golf ball and its change in velocity

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Momentum and Impulse The linear momentum p of an object of mass m moving with velocity v is defined as p = mv Same direction as velocity Units ○ kg x m / s m v p

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Momentum Vector quantity, the direction of the momentum is the same as the velocity’s Applies to two-dimensional motion as well Size of momentum: depends upon mass depends upon velocity

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Impulse In order to change the momentum of an object (say, golf ball), a force must be applied The time rate of change of momentum of an object is equal to the net force acting on it Gives an alternative statement of Newton’s second law (F Δt) is defined as the impulse Impulse is a vector quantity, the direction is the same as the direction of the force

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Graphical Interpretation of Impulse Usually force is not constant, but time- dependent If the force is not constant, use the average force applied The average force can be thought of as the constant force that would give the same impulse to the object in the time interval as the actual time-varying force gives in the interval If force is constant: impulse = F t

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Example: Impulse Applied to Auto Collisions The most important factor is the collision time or the time it takes the person to come to a rest This will reduce the chance of dying in a car crash Ways to increase the time Seat belts Air bags The air bag increases the time of the collision and absorbs some of the energy from the body

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ConcepTest Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare? 1. It takes less time to stop the ping-pong ball. 2. Both take the same time. 3. It takes more time to stop the ping-pong ball.

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ConcepTest Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare? 1. It takes less time to stop the ping-pong ball. 2. Both take the same time. 3. It takes more time to stop the ping-pong ball. Note: Because force equals the time rate of change of momentum, the two balls loose momentum at the same rate. If both balls initially had the same momenta, it takes the same amount of time to stop them.

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Problem: Teeing Off A 50-g golf ball at rest is hit by “Big Bertha” club with 500-g mass. After the collision, golf leaves with velocity of 50 m/s. a)Find impulse imparted to ball b)Assuming club in contact with ball for 0.5 ms, find average force acting on golf ball

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Problem: teeing off Given: mass: m=50 g = 0.050 kg velocity: v=50 m/s Find: impulse=? F average =? 1. Use impulse-momentum relation: 2. Having found impulse, find the average force from the definition of impulse: Note: according to Newton’s 3 rd law, that is also a reaction force to club hitting the ball:

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Momentum and Impulse Can describe Newton’s laws more precisely with momentum Law I : If no forces act, F net = 0 so a = 0 which implies that velocity is constant p = mv, therefore momentum is constant Momentum constant => conserved quantity Law II : F = Δp / Δt = m (Δv / Δt) = ma Law III : Rate of change of momentum generated by action force is exactly opposite to the rate of change of momentum by the reaction force

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Conservation of Momentum DDefinition: an isolated system is the one that has no external forces acting on it A collision may be the result of physical contact between two objects “Contact” may also arise from the electrostatic interactions of the electrons in the surface atoms of the bodies Momentum in an isolated system in which a collision occurs is conserved (regardless of the nature of the forces between the objects)

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Conservation of Momentum The principle of conservation of momentum states when no external forces act on a system consisting of two objects that collide with each other, the total momentum of the system before the collision is equal to the total momentum of the system after the collision

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Conservation of Momentum Mathematically: Momentum is conserved for the system of objects The system includes all the objects interacting with each other Assumes only internal forces are acting during the collision Can be generalized to any number of objects

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Momentum and Impulse Example:

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Conservation of Momentum If the isolated system consists of two objects undergoing a collision, the total momentum of the system is the same before and after the collision. Same concept as energy m 1 v 1i + m 2 v 2i = m 1 v 1f + m 2 v 2f

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Conservation of Momentum Remember conservation of momentum applies to the system The system includes all the objects interacting with each other Assumes only internal forces are acting during the collision You must define the isolated system Can be generalized to any number of objects

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ConcepTest 7.1 Rolling in the Rain 1) speeds up 2) maintains constant speed 3) slows down 4) stops immediately 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)

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increases decrease Since the rain falls in vertically, it adds no momentum to the box, thus the box’s momentum is conserved. However, since the mass of the box slowly increases with the added rain, its velocity has to decrease. ConcepTest 7.1 Rolling in the Rain 1) speeds up 2) maintains constant speed 3) slows down 4) stops immediately 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) Follow-up: What happens to the cart when it stops raining?

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Collisions Momentum is conserved in any collision We can define two types of collisions to describe how momentum is transferred within the system Elastic ○ Both momentum and kinetic energy are conserved Inelastic ○ Kinetic energy is not conserved, momentum is conserved

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Collisions Elastic Collisions Momentum is conserved Total Energy is conserved

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Collisions Elastic Collisions Instead of using the energy quadratic equation, we can simplify the equation to The relative velocity of the two objects before the collision equals the negative of the relative velocity of the two objects after the collision

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Collisions Elastic Collision Consider the case: both targets moving So, m 1 v 1i + m 2 v 2i = m 1 v 1f + m 2 v 2f ½ m 1 v 2 1i + ½ m 2 v 2 2i = ½ m 1 v 2 1f + ½m 2 v 2 We know m 1, m 2, v 1i, and v 2i Solve for v 1f, v 2f

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Collisions General Equation v 1f = ————— v 1i + ————— v 2i v 2f = ————— v 1i + ————— v 2i m 1 – m 2 2m 2 2m 2 m 1 – m 2 m 1 +m 2 m 1 + m 2

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Collisions Consider the case v 2i = 0 we get, v 1f = ———— v 1i v 2f = ———— v 1i m 1 – m 2 2m 2 m 1 + m 2

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Collisions (special cases) Special situations 1. Equal masses (m 2 = m 1 ) Exchange velocities If v 2i = 0, v 1f = 0 and v 2f = v 2i ○ body 1 stops and body 2 takes off with speed v 1i

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Collisions (special cases) 1. Massive target (m 2 >> m 1 ) v 1f ≈ -v 1i and v 2f ≈ (2m 1 / m 2 )v 1i body 1 bounces back in opposite direction, body 2 moves slowly ○ Example: golf ball at cannonball

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Collisions (special cases) 1. Massive projectile (m 1 >> m 2 ) v 1f ≈ v 1i and v 2f ≈ 2v 1i Body 1 continues at same speed, body two at twice the speed! ○ Example: cannonball at golf ball

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Collisions Example

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Types of Collisions MMomentum is conserved in any collision what about kinetic energy? IInelastic collisions Kinetic energy is not conserved ○S○Some of the kinetic energy is converted into other types of energy such as heat, sound, work to permanently deform an object Perfectly inelastic collisions occur when the objects stick together ○N○Not all of the KE is necessarily lost AActual collisions Most collisions fall between elastic and perfectly inelastic collisions

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Collisions Perfectly Inelastic Objects stick together ○ Not all of the KE is necessarily lost Most real life collisions fall between elastic and perfectly inelastic collisions

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Collisions Perfectly Inelastic Collisions Since both masses have the same velocity, conservation of momentum becomes, m 1 v 1i + m 2 v 2i = (m 1 + m 2 )v f

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Collisions Example of perfectly inelastic collision Momentum is conserved, kinetic energy is not conserved and converted into heat, sound, and destruction of smaller car

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Collisions Application – Perfectly Inelastic Collision Ballistic Pendulum ○ Initially used to measure speeds of bullets before electronic timing devices were developed.

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Sketches for Collision Problems Draw “before” and “after” sketches Label each object include the direction of velocity keep track of subscripts

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Sketches for Perfectly Inelastic Collisions The objects stick together Include all the velocity directions The “after” collision combines the masses

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Perfectly Inelastic Collisions: WWhen two objects stick together after the collision, they have undergone a perfectly inelastic collision SSuppose, for example, v 2i =0. Conservation of momentum becomes

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WWhat amount of KE lost during collision? lost in heat/”gluing”/sound/…

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Collisions Example – Perfectly Inelastic Collision Two objects, one with mass of 5kg and the other 3kg, approach each other with velocities of 2m/s from opposite directions. After they collide, they stick together. What is the initial and final kinetic energy? M = 5 kg M = 3 kg

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Glancing Collisions For a general collision of two objects in three-dimensional space, the conservation of momentum principle implies that the total momentum of the system in each direction is conserved Use subscripts for identifying the object, initial and final, and components

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Glancing Collisions The “after” velocities have x and y components Momentum is conserved in the x direction and in the y direction Apply separately to each direction

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Two-dimensional Collisions For a general collision of two objects in three- dimensional space, the conservation of momentum principle … implies that the total momentum of the system in each direction is conserved Use subscripts for identifying the object, initial and final, and components

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Glancing Collisions Two dimensional collisions Momentum is still conserved Apply vector components to conservation of momentum equation Applications Billiard balls Particle physics

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Center of Mass In (a), the diver’s motion is pure translation; in (b) it is translation plus rotation. There is one point that moves in the same path a particle would take if subjected to the same force as the diver. This point is called the center of mass (CM).

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Center of Mass The general motion of an object can be considered as the sum of the translational motion of the CM, plus rotational, vibrational, or other forms of motion about the CM.

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Center of Mass For two particles, the center of mass lies closer to the one with the most mass where M is the total mass.

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Center of Mass Example

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