Chapter 7 Linear Momentum

Momentum is a very useful quantity for the analysis of motion of objects. It is calculated by multiplying the mass and velocity for an object. The symbol for momentum is p.

p = mv The units of momentum are kg m/sec. The form of Newton’s Second Law that we have been using is actually a special case of a more general law. F_net = delta p/ delta t

The text shows how F = ma is derived from the general law.

*1. TRUE or FALSE F = ma is the form of the original "Newton’s Second Law."

*2. TRUE or FALSE The equation F = ma is useful for doing calculations on a rocket which is lifting off a launch pad.

In a system that consists of more than one object, the total momentum is found by adding the separate momenta of each object. If the external forces add to zero then the total momentum of the system will be conserved (p_total will be a constant).

The total momentum of an isolated system of bodies remains constant.

A halfback who weighs 180 pounds launches himself towards the goal line with the football at 10 m/sec. A 220 pound linebacker launches himself towards the halfback at 7 m/sec. They collide such that the football is just a fraction of an inch away from the goal line. Will the ball cross the goal line?

Questions for thought:

3. Suppose that a car crash on 10^th street occurs and both drivers apply their brakes during the collision. Is momentum conserved for this collision? Explain.

4. Why does a rifle recoil when it is shot?

Impulse is found by multiplying the average force by the length of time that the force was acting. Impulse is equal to the change in momentum. F Δt = Δp

5. Two objects are thrown with a speed of 5 m/s toward a wall. Ball A stops when it hits the wall and does not rebound. Ball B rebounds. TRUE or FALSE The impulse on Ball A is greater than the impulse on Ball B. You may assume the contact times are equal for the two collisions.

*6. TRUE or FALSE The use of "shock absorbers" in car bumpers is just a marketing gimmick and has no practical benefit of reducing the damage to a car in a 5 mile/hour accident.

7. Why is it important to bend your knees as you land after jumping?

8. Describe the hitting technique that causes the greatest change in momentum for a tennis ball.

*There are three types of collisions that occur between objects:

1) elastic (also called perfectly elastic) KE is conserved eg. atomic collisions;

2) inelastic some KE is lost e.g. kicking a football, hitting a tennis ball;

3) completely inelastic much KE is lost e.g. a lineman tackles a halfback in football; two train cars link together; air track cars with velcro strips stick together; a ball is caught by a pendulum.

*If the external forces can be ignored, momentum is conserved in all types of collisions. This fact makes the momentum quantity very useful in analyzing collisions.

*9. True or False Conservation of momentum cannot be used in elastic collisions.

10. True or False In an inelastic collision, the direction of the velocity of the combined object will be the same as the initial velocity of the object which is more massive.

*11. True or False In a collision between a car and a truck, the force of the car on the truck is less than the force of the truck on the car.

When the equations of conservation of momentum and conservation of kinetic energy are combined we find that

How do these equations simplify for the case of V₂ = 0 and

a) m₁ = m₂ ? b) m₁ >> m₂ ? c) m₁ << m₂ ?

Some groups will do the following demonstration and report results to the class. Lay a marble in the groove of a plastic ruler (or just on the table).

a) Shoot a marble with similar mass at the stationary marble. Observe the collision.

b) Shoot a steel ball at a stationary marble. Observe the collision.

c) Shoot a marble at stationary steel ball. Observe the collision.

12. True or False In an inelastic collision, kinetic energy is conserved.

13. True or False In an elastic collision the internal forces are zero.

14. True or False In an elastic collision the velocities of all objects remain constant.

15. True or False In an inelastic collision the value of the total kinetic energy in the system after the collision occurs is always less than the total kinetic energy in the system before the collision occurs.

In example 7-9 there is friction present when the bullet enters the wood. Why is it allowed to use the conservation of energy principle without including a term that represents the work done by friction as one step in the solution?

Why is momentum not conserved as the pendulum swings upward after the collision?

*Momentum is conserved separately in both the x and y directions. If a 2 dimensional velocity is present you must find the x and y components and calculate momenta separately for the x component and then for the y component.

A lump of clay (4.3 kg) is moving horizontally through the air with a speed of 7.8 m/s. A second lump of clay (5.6 kg) is moving at 10.2 m/s at right angles to the first lump such that a collision occurs. The two lumps stick together. What is the speed and direction of the combined object.

We will work in groups on the following problem: Suppose two billiard balls (each with mass = 0.3 kg ) collide. Ball 1 is moving at 5 m/s along the positive X axis. Ball 2 is at rest at X = 30 cm. After the collision ball 1 is moving at an 80 degree angle to the positive X axis. Find the magnitude of the velocity of ball 1 and 2 and the angle of the velocity of ball 2. There are 3 unknowns. How many equations should you write down before you attempt to find an answer?

The solution will take a full sheet of paper.

Note: The physics analysis of an explosion is very similar to the physics analysis of a collision. An explosion is treated as a collision that is seen in reverse time sequence.

16. You are standing in the middle of a lake of ice that has a coefficient of friction of zero. How will you get off of the ice before it melts in the spring?

We will study rotation of extended objects (e.g. discs, human bodies ) in chapter 8. In preparation for this the book discusses center of mass. The center of mass is the location in the object that moves in the same path as a point particle of the same mass would move if the same net force acted on the point particle. The red plus sign on the wrench in Figure 7-20 is the center of mass location for the wrench. The wrench is sliding and rotating on a low friction surface.

17. A bomb is launched with a slingshot at an angle of 50 degrees as shown below. Due to a faulty fuse the bomb blows up in midair. Sketch the approximate location of the "middle" of the bomb debris on the ground.

What is the Center of Gravity?

7.9 and 7.10 Read these but we won’t discuss them in class unless someone has a question.

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