Momentum and Collisions Objectives To compare the momentum of different objects. To compare the momentum of the same object moving with different velocities. To identify examples of change in the momentum of an object. To describe changes in momentum in terms of force and time. Momentum The linear momentum pof an object of mass m moving with a velocity is defined as the product of the mass and the velocity v ◦ p mv ◦ SI Units are kg m / s ◦ Vector quantity, the direction of the momentum is the same as the velocity’s Momentum components p x mv x and p y mv y Applies to two-dimensional motion Momentum for any object at rest = ______________________ Double the mass results in ________________ the momentum Quadruple the velocity results in ______________ the momentum. Determine the momentum of a … ◦ 60 kg halfback moving eastward at 9 m/s ◦ 1000 kg car moving northward at 20 m/s ◦ 40 kg freshman moving southward at 2 m/s A car possesses 20 000 units of momentum. What would be the car’s new momentum if … ◦ Its velocity were doubled ◦ Its velocity were tripled ◦ Its mass were doubled (by adding more passengers and a greater load) ◦ Both its velocity and mass were doubled Impulse In order to change the momentum of an object, a force must be applied The rate of change of momentum of an object is equal to the net force acting on it p m(vf vi ) ◦ Fnet t t ◦ Gives an alternative statement of Newton’s second law (F=ma, where a = (vf-vi)/∆t ) Impulse cont. When a single, constant force acts on the object, there is an impulse delivered to the object ◦ I Ft ◦ I is defined as the impulse ◦ Vector quantity, the direction is the same as the direction of the force Impulse-Momentum Theorem The theorem states that the impulse acting on the object is equal to the change in momentum of the object ◦ Ft p mv f mv i ◦ If the force is not constant, use the average force applied Average Force in Impulse 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 Average Force cont. The impulse imparted by a force during the time interval Δt is equal to the area under the force-time graph from the beginning to the end of the time interval Or, the impulse is equal to the average force multiplied by the time interval, Fav t p Which of the above has the greatest velocity change? The greatest acceleration? The greatest momentum change? The greatest impulse? Which of the above has the greatest velocity change? The greatest acceleration? The greatest momentum change? The greatest impulse? A 1400 kg car moving westward with a velocity of 15 m/s collides with a utility pole and is brought to rest in 0.30 s. Find the magnitude of the force exerted on the car during the collision. A 2250 kg car traveling to the west slows down uniformly from 20.0 m/s to 5.0 m/s. How long does it take the car to decelerate if the force on the car is 8450 N to the east? How far does the car travel during the deceleration? If the maximum coefficient of kinetic friction between a 2300 kg car and road is 0.50, what is the minimum stopping distance for a car moving at 29 m/s? 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 Air Bags The air bag increases the time of the collision It will also absorb some of the energy from the body It will spread out the area of contact ◦ decreases the pressure ◦ helps prevent penetration wounds Review In order to change the momentum of an object a ________ must be applied. How can the magnitude of an applied force be reduced when an object undergoes a change in momentum? Conservation of Momentum Momentum in an isolated system in which a collision occurs is conserved ◦ 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 ◦ An isolated system will have no external forces Conservation of Momentum, cont 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 remains constant in time ◦ Specifically, the total momentum before the collision will equal the total momentum after the collision Conservation of Momentum, cont. Mathematically: m1v1i m2v2i m1v1f m2v2f ◦ 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 Think Pair Share A boy standing at one end of a floating raft that is stationary relative to the shore walks to the opposite end of the raft, away from the shore. As a consequence, the raft (a) remains stationary, (b) moves away from the shore, or (c) moves toward the shore. (Hint: Use conservation of momentum.) A 76 kg boater, initially at rest in a stationary 45 kg boat, steps out of the boat and onto the dock. If the boater moves out of the boat with a velocity of 2.5 m/s to the right, what is the final velocity of the boat? A baseball player wishes to maintain his physical condition during the winter. He uses a 50.0 kg pitching machine to help him, placing the machine on the pitcher’s mound. The ground is covered with a thin layer of ice so that there is negligible friction between the ground and the machine. The machine fires a 0.15 kg baseball horizontally with a speed of 36 m/s. What is the recoil speed of the machine? Types of Collisions Momentum is conserved in any collision Inelastic collisions ◦ Kinetic energy is not conserved Some of the kinetic energy is converted into other types of energy such as heat, sound, work to permanently deform an object Example: rubber ball with hard floor ◦ Perfectly inelastic collisions occur when the objects stick together Not all of the KE is necessarily lost Example: two pieces of putty, meteorite collides with the earth More Types of Collisions Elastic collision ◦ both momentum and kinetic energy are conserved ◦ Example: billiard balls, air molecules with the walls of a container Actual collisions ◦ Most collisions fall between elastic and perfectly inelastic collisions Think Pair Share A car and a large truck traveling at the same speed collide head-on and stick together. Which vehicle experiences the larger change in the magnitude of its momentum? (a) the car (b) the truck (c) the change in the magnitude of momentum is the same for both (d) impossible to determine More About Perfectly Inelastic Collisions When two objects stick together after the collision, they have undergone a perfectly inelastic collision Conservation of momentum becomes m1v1i m2v 2i (m1 m2 )v f Some General Notes About Collisions Momentum is a vector quantity ◦ Direction is important ◦ Be sure to have the correct signs An object of mass m moves to the right with a speed v. It collides head-on with an object of mass 3m moving with speed v/3 in the opposite direction. If the two objects stick together, what is the speed of the combined object, of mass 4m, after the collision? (a) 0 (b) v/2 (c) v (d) 2v A skater is using very low friction rollerblades. A friend throws a Frisbee® at her, on the straight line along which she is coasting. Describe each of the following events as an elastic, an inelastic, or a perfectly inelastic collision between the skater and the Frisbee: (a) She catches the Frisbee and holds it. (b) She tries to catch the Frisbee, but it bounces off her hands and falls to the ground in front of her. (c) She catches the Frisbee and immediately throws it back with the same speed (relative to the ground) to her friend. In a perfectly inelastic one-dimensional collision between two objects, what condition alone is necessary so that all of the original kinetic energy of the system is gone after the collision? (a) The objects must have momenta with the same magnitude but opposite directions. (b) The objects must have the same mass. (c) The objects must have the same velocity. (d) The objects must have the same speed, with velocity vectors in opposite directions. An empty train car moving east at 21 m/s collides with a loaded train car initially at rest that has twice the mass of the empty car. The two cars stick together. A) Find the velocity of the cars after the collision. B) Find the final speed if the loaded car moving at 17 m/s had hit the empty car initially at rest. An empty train car moving at 15 m/s collides with a loaded car of three times the mass moving at one-third the speed of the empty car. The cars stick together. Find the speed of the cars after the collision. A clay ball with a mass of 0.35 kg hits another 0.35 kg ball at rest, and the two stick together. The first ball has an initial speed of 4.2 m/s. What is the final speed of the balls? Calculate the decrease in kinetic energy that occurs during the collision. An 1800 kg car stopped at a traffic light is struck from the rear by a 900kg car and the two become entangled. If the smaller car was moving at 20 m/s before the collision, what is the speed of the entangled cars after the collision? Two balls of mud collide head on in a perfectly inelastic collision. Suppose m1 = 0.500 kg and m2 = 0.250 kg, v1i = 4 m/s and v2i = -3.00 m/s. Find the velocity of the composite ball of mud after the collision. What is the change in kinetic energy of the system after the collision? More About Elastic Collisions Both momentum and kinetic energy are conserved Typically have two unknowns m1v1i m2 v 2i m1v1f m2 v 2 f 1 1 1 1 Solve 1i m2 v 2i m1v1f m2 2 m1vthe equations simultaneouslyv 2 f 2 2 2 2 2 2 2 Elastic Collisions, cont. A simpler equation can be used in place of the KE equation v1i v 2i (v1f v 2f ) Two billiard balls each with a mass of 0.35 kg, strike each other head on elastically. One ball is initially moving left at 4.1 m/s and ends up moving right at 3.5 m/s. The second ball is initially moving to the right at 3.5 m/s. Find the final velocity of the second ball. Two nonrotating balls on a frictionless surface collide elastically head on. The first ball has a mass of 15 g and an initial velocity of 3.5 m/s to the right, while the second ball has a mass of 22 g and an initial velocity of 4.0 m/s to the left. The final velocity of the 15 g ball is 5.4 m/s to the left. What is the final velocity of the 22 g ball? Two billiard balls move toward one another. The balls have identical masses and assume that the collision between them is elastic. If the initial velocities of the balls are +30.0 cm/s and -20.0 cm/s, what is the velocity of each ball after the collision? Find the final velocity of the two balls if the ball with initial velocity -20 cm/s has a mass equal to half that of the ball with initial velocity of +30 cm/s. Summary of Types of Collisions In an elastic collision, both momentum and kinetic energy are conserved In an inelastic collision, momentum is conserved but kinetic energy is not In a perfectly inelastic collision, momentum is conserved, kinetic energy is not, and the two objects stick together after the collision, so their final velocities are the same Sketches for Collision Problems Draw “before” and “after” sketches Label each object ◦ include the direction of velocity ◦ keep track of subscripts Sketches for Perfectly Inelastic Collisions The objects stick together Include all the velocity directions The “after” collision combines the masses 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 ◦ m1v1ix m2 v 2ix m1v1f x m2 v 2 f x and Use m2 v 2iy m1v1f y m v 2 ◦m1v1iysubscripts for identifying 2thef yobject, initial and final velocities, and components Glancing Collisions The “after” velocities have x and y components Momentum is conserved in the x direction and in the y direction Apply conservation of momentum separately to each direction Problem Solving for Two- Dimensional Collisions Coordinates: Set up coordinate axes and define your velocities with respect to these axes ◦ It is convenient to choose the x- or y- axis to coincide with one of the initial velocities Draw: In your sketch, draw and label all the velocities and masses Problem Solving for Two- Dimensional Collisions, 2 Conservation of Momentum: Write expressions for the x and y components of the momentum of each object before and after the collision Write expressions for the total momentum before and after the collision in the x-direction and in the y-direction Problem Solving for Two- Dimensional Collisions, 3 Conservation of Energy: If the collision is elastic, write an expression for the total energy before and after the collision ◦ Equate the two expressions ◦ Fill in the known values ◦ Solve the quadratic equations Can’t be simplified Problem Solving for Two- Dimensional Collisions, 4 Solve for the unknown quantities ◦ Solve the equations simultaneously ◦ There will be two equations for inelastic collisions ◦ There will be three equations for elastic collisions A billiard ball moving with speed 3.0 m/s in the +x direction strikes an equal mass ball initially at rest . The two balls are observed to move off at 45o, ball 1 above the x axis and ball 2 below. What are the speeds of the two balls? Lab Momentum, Energy and Collisions Individually answer the preliminary questions on a sheet of paper. When completed get my initials on your paper. Staple this to your lab report when you turn it in. You will each type and turn in your data table and analysis questions with prelim questions attached. .
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