11.1 Kinetic and Potential Energy

This chapter discusses energy. Energy is an amazing concept. It is almost like magic because we can't always see it, just it's effects and manifestations. For example, consider something as simple as a baseball that was just pitched.

• Why does it keep going through the air, even though the pitcher is no longer applying force to it?
• What did the pitcher give it that keeps it going?
• Just what, exactly, does that ball have that it didn't have when it was in the pitcher's hand?
• It has kinetic energy! But where did that energy come from?
• How did it get into the ball if it wasn't there before?

I want you to figure out what the ball has received that it manages to keep, even after it has left the pitcher's hand!

The explanations and answers are tricky and sometimes nebulous as we still don't fully understand how all energy behaves. We can look at some of the simpler concepts though, and give ourselves a little better understanding of how our lives work! So, here we go.. hold on to your pencils!

Let's return to the baseball thing. What did the pitcher do TO the ball?

• He applied a force to it which caused it to change its motion (acceleration).
• There must have been some time involved as he applied the force, so we can also say that he applied an impulse to the ball, and therefore it's momentum changed.
• Further, there was some distance over which the force was applied ("....there's the wind-up, here comes the pitch..." etc.)so we can infer that WORK was done on the ball.

Take a look at some of these formulas that apply:

We remember the impulse-momentum theorem as Ft = mv , so

(1.)

The work done on the ball is W = Fd, but we just saw that so substituting into the work formula we get

(2.)

This stuff all seems ok. There is work done on the ball, and there was an impulse Ft. But that was all before the ball left the pitchers hand. I want to know what the ball has AFTER it leaves the pitcher's hand that allows it to keep going.

Check out that last work formula (2.) again.

• The distance (d) was the distance involved while the pitcher was applying force, in other words, while it was still in the pitcher's hand.
• I don't want to consider that in my quest.. so lets see if it really needs to be there. I'm going after "d" here...

In that same work formula, what values are given on the right side of the equation that the ball will STILL have after the ball leave's the pitcher's hand?

• It will still have its MASS and whatever FINAL VELOCITY it attained just at the end of the pitch.
• So.. how come the distance thing is still there. Actually, it doesn't need to be. Check this:

When something accelerates (like the ball in the pitcher's hand), it's velocity changes according to the formula

(3.)

If the ball starts from zero velocity (v_i) and ends up travelling at some final velocity (v_f) just as it leaves the hand, then

(4. ) because v_i was zero. And,

(5.) .

These values d and a in equation (5.) all have to do with what is going on in the pitcher's hand except the final velocity v_f

Watch this:

substitute for "d" in the work formula (2.) and you get

That got rid of the distance! I still don't want that acceleration in my formula, because I want to examine only those properties the ball has after it leaves the pitcher's hand, and acceleration happened IN the pitcher's hand. BUT.. that v/t in the formula.. what;s that? That's also the acceleration! so..

and the a's cancel out.. which is cool, because after the ball leaves the pitcher's hand, it shouldn't be accelerating anymore anyway! The ball is now clear of the acceleration and this leaves us with

, or

THE MAIN POINT IS RIGHT HERE:

• In this formula, we see only what work gives an object that it can keep! Mass and it's final velocity.
• This value, , is that which an object has after work was done on it! We made it. This value is called Kinetic Energy, and is denoted as KE

Doing work on the ball has given it kinetic energy, and if you followed the derivation,

• the amount of kinetic energy aquired is equivalent mathematically to the work done on it.
• This is called the work-energy theorem, and can be simply expressed as

You try: do practice problems 1-3 on page 251

Potential Energy

• Only changes in energy can be determined.
• Potential energy may be positive or negative, depending on the position of the object relative to the reference level
• (Kinetic energy is always positive)

Gravitational Potential Energy

• If an object moves away from the Earth, energy is stored in the system as a result of the gravitational interaction between the object and the Earth.
• Formula: Ug = mgh
• h is always measured relative to a reference level at which Ug is defined as zero. The reference level may be determined as any convenient location.
  

Elastic Potential Energy

• If an object moves away from the Earth, energy is stored in the system as a result of the gravitational interaction between the object and the Earth.
• Formula: Ug = mgh
• Potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring.
• It is equal to the work done to the stretch the spring, which depends upon the spring constant 'k' as well as the distance stretched 'x'.
• F = -k x
• 

Set #2. Do problems 38-47

Set #3 Do problems 48-56