Ying Yi PhD Chapter 6 Momentum and Collisions 1 PHYS HCC

Outline PHYS HCC 2 Momentum and Impulse Conservation of Momentum Collision  Inelastic  Elastic

Momentum Definition: The linear momentum of an object of mass m moving with a velocity is defined as the product of the mass and the velocity SI Units: kgm / s Note that: Vector quantity, the direction of the momentum is the same as the velocity’s 3 PHYS HCC

Components of Momentum X components: p x = m v x Y components: p y = m v y Momentum is related to kinetic energy 4 PHYS HCC

Momentum and Kinetic Energy PHYS HCC 5 Definition: Units: Relation: kgm / s J

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 Gives an alternative statement of Newton’s second law 6 PHYS HCC

Impulse When a single, constant force acts on the object, there is an impulse delivered to the object is defined as the impulse Vector quantity, the direction is the same as the direction of the force 7 PHYS HCC

Impulse-Momentum Theorem The theorem states that the impulse acting on the object is equal to the change in momentum of the object If the force is not constant, use the average force applied 8 PHYS HCC

9 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 Δ t 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, 10 PHYS 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 HCC

12 Typical Collision Values For a 75 kg person traveling at 27 m/s and coming to stop in s F = -2.0 x 10 5 N a = 280 g Almost certainly fatal

Comparison of Accelerations PHYS HCC 13 About 4g About 2g 280g

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 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 HCC

Example 6.1 Teeing off PHYS HCC 16 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.)

Group Problem: How good are the Bumpers? PHYS HCC 17 In a crash test, a car of mass 1.5×10 3 kg collides with a wall and rebounds as in Figure 6.4a. The initial and final velocities of the car are v i =-15.0 m/s and v f =2.60 m/s, respectively. If the collision lasts for s, find (a) the impulse delivered to the car due to the collision and (b) the size and direction of the average force exerted on the car.

Conservation of Momentum PHYS HCC 18 Newton’s Third Law

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 19 PHYS HCC

Example 6.3 The Archer PHYS HCC 20 An archer stands at rest on frictionless ice; his total mass including his bow and quiver of arrows is kg. (a) If the archer fires a kg arrow horizontally at 50.0 m/s in the positive x-direction, what is his subsequent velocity across the ice? (b) He then fires a second identical arrow at the same speed relative to the ground but at an angle of 30.0° above the horizontal. Find his new speed. (c) Estimate the average normal force acting on the archer as the second arrow is accelerated by the bowstring. Assume a draw length of m.

PHYS HCC 21 Forces in a Collision The force with which object 1 acts on object 2 is equal and opposite to the force with which object 2 acts on object 1 Impulses are also equal and opposite

Types of collisions (Momentum is conserved) PHYS HCC 22 Inelastic Collision Elastic 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 Both momentum and kinetic energy are conserved

Collision Examples PHYS HCC 23

PHYS HCC 24 Perfectly Inelastic Collisions When two objects stick together after the collision, they have undergone a perfectly inelastic collision Conservation of momentum becomes

Elastic Collisions Both momentum and kinetic energy are conserved Typically have two unknowns Solve the equations simultaneously 25 PHYS HCC

26 Elastic Collisions, cont. A simpler equation can be used in place of the KE equation

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 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 HCC

Example 6.5: The Ballistic Pendulum PHYS HCC 29 The ballistic pendulum is a device used to measure the speed of a fast-moving projectile such as a bullet. The bullet is fired into a large block of wood suspended from some light wires. The bullet is topped by the block, and the entire system swings up to a height h. It is possible of obtain the initial speed of the bullet by measuring h and the two masses. As an example of the technique, assuming that the mass of the bullet, m 1, is 5.00 g, the mass of the pendulum, m 2, is kg, and h is 5.00 cm. (a) Find the velocity of the system after the bullet embeds in the block. (b) Calculate the initial speed of the bullet.

Group Problem: A truck versus a compact PHYS HCC 30 A pickup truck with mass 1.80×10 3 kg is traveling eastbound at m/s, while a compact car with mass 9.00×10 2 kg is traveling west bound at 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.

PHYS HCC 31 Sketches for Collision Problems Draw “before” and “after” sketches Label each object Include the direction of velocity Keep track of subscripts

PHYS HCC 32 Sketches for Perfectly Inelastic Collisions The objects stick together Include all the velocity directions The “after” collision combines the masses Both move with the same velocity

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 velocities, and components 33 PHYS HCC

34 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 Diagram: In your sketch, draw and label all the velocities and masses 35 PHYS HCC

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 36 PHYS HCC

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 37 PHYS HCC

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 38 PHYS HCC

Example 6.8 collision at an intersection PHYS HCC 39 A car with mass 1.50×10 3 kg traveling east at a speed of 25.0 m/s collides at an intersection with a 2.50×10 3 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.

PHYS HCC 40 Rocket Propulsion The rocket is accelerated as a result of the thrust of the exhaust gases

Thank you PHYS HCC 41