#### Transcript Conservation of Momentum

Chapter 7 Impulse and Momentum Ying Yi PhD 1 PHYS I @ HCC Outline Impulse and Momentum Theorem Conservation of Momentum Collision Inelastic Elastic 2 PHYS I @ HCC Momentum Definition: The linear momentum p of 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: kg•m / s Note that: Vector quantity, the direction of the momentum is the same as the velocity’s 3 PHYS I @ HCC Components of Momentum X components: px = m vx Y components: p y = m vy Momentum is related to kinetic energy p2 KE 2m 4 PHYS I @ HCC Momentum and Kinetic Energy Definition: Units: Relation: 5 PHYS I @ HCC p mv KE 1 mv 2 2 J kg•m / s p2 KE 2m Change of Momentum and Force In order to change the momentum of an object, a force must be applied The time rate of change of momentum of an object is equal to the net force acting on it p m(v f v i ) Fnet t t Gives an alternative statement of Newton’s second law 6 PHYS I @ HCC Impulse 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 7 PHYS I @ HCC Impulse-Momentum Theorem The theorem states that the impulse acting on the object is equal to the change in momentum of the object I Ft p mvf mvi If the force is not constant, use the average force applied 8 PHYS I @ HCC 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 9 PHYS I @ HCC 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 10 PHYS I @ HCC 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 11 PHYS I @ HCC Typical Collision Values For a 75 kg person traveling at 27 m/s and coming to stop in 0.010 s F = -2.0 x 105 N a = 280 g Almost certainly fatal 12 PHYS I @ HCC Comparison of Accelerations About 2g 280g About 4g 13 PHYS I @ HCC Survival Increase time Seat belt Restrain people so it takes more time for them to stop New time is about 0.15 seconds 14 PHYS I @ HCC 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 15 PHYS I @ HCC Example 7.1 A well hit ball A baseball (m=0.14 kg) has an initial velocity of V0=-38 m/s as it approaches a bat. We have chosen the direction of approach as the negative direction. The bat applies an average force F that is much larger that the weight of the ball, and the ball departs from the bat with a final velocity of Vf=+58 m/s. (a) Determine the impulse applied to the ball by the bat. (b) Assuming that the time of the contact is ∆t=1.6×10-3 s, find the average force exerted on the ball by the bat. 16 PHYS I @ HCC Group Problem: Teeing off A golf ball with mass 5.0×10-2 kg is struck with a club as in Figure 6.3. The force on the ball varies from zero when contact is made up to some maximum value and then back to zero when the ball leaves the club, as in the graph of force vs. time in Figure 6.1. Assume that the ball leaves the club face with a velocity of +44 m/s. (a) Find the magnitude of the impulse due to the collision. (b) Estimate the duration of the collision and the average force acting on the ball. (Assume contacting distance is 2cm.) 17 PHYS I @ HCC Conservation of Momentum （W 1 F 12）t m1V f 1 m1V i1 （W2 F 21）t m2V f 2 m2V i 2 Newton’s Third Law F 21 F12 m1V i1 m2V i 2 m1V f 1 m2V f 2 18 PHYS I @ HCC How about explosion? Momentum conserved? 19 PHYS I @ HCC 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 20 PHYS I @ HCC Example 7.5 Assembling a Freight Train A freight train is being assembled in a switching yard, and Figure 7.8 shows two boxcars. Car 1 has a mass of m1=65×103 kg and moves at a velocity of v01=+0.80 m/s. Car 2, with a mass of m2=92×103 kg and a velocity of v02=+1.3 m/s, overtakes car 1 and couples to it Neglecting friction, find the common velocity vf of the cars after they become coupled. 21 PHYS I @ HCC Group Problem: Ice Skaters Starting from rest, two skaters push off against each other on smooth level ice, where friction is negligible. One is woman (m1=54 kg), and one is man (m2=88 kg). Part b of the drawing shows that the woman moves away with a velocity of vf1=+2.5 m/s. Find the recoil velocity of vf2 of the man 22 PHYS I @ HCC Types of collisions 23 Inelastic Collision 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. Perfectly inelastic collisions occur when the objects stick together Elastic Collision Both momentum and kinetic energy are conserved PHYS I @ HCC Collision Examples 24 PHYS I @ HCC Perfectly Inelastic Collisions When two objects stick together after the collision, they have undergone a perfectly inelastic collision Conservation of momentum becomes m1 v i1 m2 v i 2 (m1 m2 )v f 25 PHYS I @ HCC Elastic Collisions Both momentum and kinetic energy are conserved Typically have two unknowns m1 v i1 m2 v i 2 m1 v f 1 m2 v f 2 1 1 1 1 2 2 2 m1vi1 m2 vi 2 m1v f 1 m2v 2f 2 2 2 2 2 Solve the equations simultaneously 26 PHYS I @ HCC 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 27 PHYS I @ HCC Problem Solving for One -Dimensional Collisions Coordinates: Set up a coordinate axis and define the velocities with respect to this axis Diagram: Draw all the velocity vectors and label the velocities and the masses Conservation of Momentum: Write a general expression for the total momentum of the system before and after the collision Conservation of Energy: If the collision is elastic, write a second equation for conservation of KE, or the alternative equation (perfectly elastic collisions) Solve: The resulting equations simultaneously 28 PHYS I @ HCC Example 7.3 measuring the speed of a bullet The ballistic pendulum shown in Figure 7.12 a consists of a stationary 2.5 kg block of wood suspended by a wire of negligible mass. A 0.0100 kg bullet is fired into the block, and the block (with the bullet in it) swings to a maximum height of 0.650 m above the initial position. Find the speed with which the bullet is fired. (Ignore air resistance) 29 PHYS I @ HCC Group Problem: A truck versus a compact A pickup truck with mass 1.80×103 kg is traveling eastbound at +15.0 m/s, while a compact car with mass 9.00×102 kg is traveling west bound at -15.0 m/s. The vehicles collide head-on, becoming entangled. (a) Find the speed of the entangled vehicles after the collision. (b) Find the change in the velocity of each vehicle. (c) Find the change in the kinetic energy of the system consisting of both vehicles. 30 PHYS I @ HCC A collision in two dimensions 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 m1 v i1x m2 v i 2 x m1 v f 1x m2 v f 2 x m1 v i1 y m2 v i 2 y m1 v f 1 y m2 v f 2 y 31 PHYS I @ HCC Problem Solving for Two-Dimensional Collisions 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 32 PHYS I @ HCC Group Problem: collision at an intersection A car with mass 1.50×103 kg traveling east at a speed of 25.0 m/s collides at an intersection with a 2.50×103 kg van traveling north at a speed of 20.0 m/s. Find the magnitude and direction of the velocity of the wreckage after the collision, assuming that the vehicles undergo a perfectly inelastic collision and assuming that friction between the vehicles and the road can be neglected. 33 PHYS I @ HCC Homework: 4,8,12,16,17,30,38,42, Thank you 34 PHYS I @ HCC