1. Momentum and Collisions
Chapter 6
Momentum and Collisions

2. Momentum
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 are kg m / s
Vector quantity, the direction of the momentum is the same as the velocity’s

3. Momentum components
Applies to two-dimensional motion

4. Impulse
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

5. Impulse cont.
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

6. 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

7. 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

8. 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

9. Example 1
Calculate the magnitude of the linear momentum for the following cases: (a) a proton with mass 1.67 × 10–27 kg, moving with a speed of 5.00 × 106 m/s; (b) a 15.0-g bullet moving with a speed of 300 m/s; (c) a 75.0-kg sprinter running with a speed of 10.0 m/s; (d) the Earth (mass = 5.98 × 1024 kg) moving with an orbital speed equal to 2.98 × 104 m/s.

10. Example 2
A 0.10-kg ball is thrown straight up into the air with an initial speed of 15 m/s. Find the momentum of the ball (a) at its maximum height and (b) halfway to its maximum height.

11. Example 3
A 75.0-kg stuntman jumps from a balcony and falls 25.0 m before colliding with a pile of mattresses. If the mattresses are compressed 1.00 m before he is brought to rest, what is the average force exerted by the mattresses on the stuntman?

12. Example 4
A kg football is thrown toward the east with a speed of 15.0 m/s. A stationary receiver catches the ball and brings it to rest in s. (a) What is the impulse delivered to the ball as it’s caught? (b) What is the average force exerted on the receiver?

13. Example 5
A force of magnitude Fx acting in the x-direction on a 2.00-kg particle varies in time as shown in Figure P6.12. Find (a) the impulse of the force, (b) the final velocity of the particle if it is initially at rest, and (c) the final velocity of the particle if it is initially moving along the x-axis with a velocity of –2.00 m/s.

14. Example 6
A pitcher throws a 0.15-kg baseball so that it crosses home plate horizontally with a speed of 20 m/s. The ball is hit straight back at the pitcher with a final speed of 22 m/s. (a) What is the impulse delivered to the ball? (b) Find the average force exerted by the bat on the ball if the two are in contact for 2.0 × 10–3 s.

15. 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 not external forces

16. 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

17. Conservation of Momentum, cont.
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

18. Notes About A System
Remember conservation of momentum applies to the system
You must define the isolated system

19. 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
Perfectly inelastic collisions occur when the objects stick together
Not all of the KE is necessarily lost

20. More Types of Collisions
Elastic collision
both momentum and kinetic energy are conserved
Actual collisions
Most collisions fall between elastic and perfectly inelastic collisions

21. More About Perfectly Inelastic Collisions
When two objects stick together after the collision, they have undergone a perfectly inelastic collision
Conservation of momentum becomes

22. Some General Notes About Collisions
Momentum is a vector quantity
Direction is important
Be sure to have the correct signs

23. More About Elastic Collisions
Both momentum and kinetic energy are conserved
Typically have two unknowns
Solve the equations simultaneously

24. 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

25. Problem Solving for One -Dimensional Collisions
Coordinates: Set up a coordinate axis and define the velocities with respect to this axis
It is convenient to make your axis coincide with one of the initial velocities
Diagram: In your sketch, draw all the velocity vectors and label the velocities and the masses

26. Problem Solving for One -Dimensional Collisions, 2
Conservation of Momentum: Write a general expression for the total momentum of the system before and after the collision
Equate the two total momentum expressions
Fill in the known values

27. Problem Solving for One -Dimensional Collisions, 3
Conservation of Energy: If the collision is elastic, write a second equation for conservation of KE, or the alternative equation
This only applies to perfectly elastic collisions
Solve: the resulting equations simultaneously

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

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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. Example 7
A rifle with a weight of 30 N fires a 5.0-g bullet with a speed of 300 m/s. (a) Find the recoil speed of the rifle. (b) If a 700-N man holds the rifle firmly against his shoulder, find the recoil speed of the man and rifle.

37. Example 8
A 75.0-kg ice skater moving at 10.0 m/s crashes into a stationary skater of equal mass. After the collision, the two skaters move as a unit at 5.00 m/s. Suppose the average force a skater can experience without breaking a bone is N. If the impact time is s, does a bone break?

38. Example 9
A kg bullet is fired vertically at 200 m/s into a 0.15-kg baseball that is initially at rest. How high does the combined bullet and baseball rise after the collision, assuming the bullet embeds itself in the ball?

39. Example 10
A kg car traveling initially with a speed of 25.0 m/s in an easterly direction crashes into the rear end of a kg truck moving in the same direction at 20.0 m/s (Fig. P6.32). The velocity of the car right after the collision is 18.0 m/s to the east. (a) What is the velocity of the truck right after the collision? (b) How much mechanical energy is lost in the collision? Account for this loss in energy.

40. Example 11
A 12.0-g bullet is fired horizontally into a 100-g wooden block that is initially at rest on a frictionless horizontal surface and connected to a spring having spring constant 150 N/m. The bullet becomes embedded in the block. If the bullet–block system compresses the spring by a maximum of 80.0 cm, what was the speed of the bullet at impact with the block?

41. Example 12
A billiard ball rolling across a table at 1.50 m/s makes a head-on elastic collision with an identical ball. Find the speed of each ball after the collision (a) when the second ball is initially at rest, (b) when the second ball is moving toward the first at a speed of 1.00 m/s, and (c) when the second ball is moving away from the first at a speed of 1.00 m/s.

42. Example 13
A 90-kg fullback moving east with a speed of 5.0 m/s is tackled by a 95-kg opponent running north at 3.0 m/s. If the collision is perfectly inelastic, calculate (a) the velocity of the players just after the tackle and (b) the kinetic energy lost as a result of the collision. Can you account for the missing energy?