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Notes: Going Further With Energy
Level 1: Conservation of Mechanical Energy
So far, we’ve mostly studied situations where the mechanical energy of a system has stayed the same- like roller
coasters without friction and objects in free fall. This helped us learn a lot about how mechanical, potential, and kinetic
energy work. Notice, in the example below, the mechanical energy (potential + kinetic energy) stays the same the whole
way through the roller coaster.
When the mechanical energy doesn’t change, we say it is conserved.
This roller coaster shows mechanical energy being conserved.
Mechanical energy is conserved if you don’t have a force adding energy to the system or taking it away. For example,
there are no engines speeding a car up, no friction slowing it down, no jet packs making it accelerate, no air resistance
causing drag.
When the mechanical energy does change, we say it is not the energy was not
conserved.
We say energy is not conserved when a car puts on its brakes and comes to a stop, when a rocket booster speeds the
shuttle up into the sky, when friction makes a ball roll to a stop. In these situations, energy was either entering the
system or leaving it.
Level 2: Work
Let’s say we did want the mechanical energy to change- we wanted a car to speed up or we wanted to an object to slow
down. How can we change the mechanical energy? Something that changes the mechanical energy of a system is called
work.
𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 + 𝑊𝑜𝑟𝑘 = 𝐹𝑖𝑛𝑎𝑙 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦
𝑀𝐸𝑖𝑛𝑖𝑡𝑖𝑎𝑙 + 𝑊 = 𝑀𝐸𝑓𝑖𝑛𝑎𝑙
Positive Work
Lets say a child is sliding past on a sled. She has no potential energy and 400 J of kinetic energy. How much mechanical
energy does she have?
If you wrote 400 J, that’s right. Now, imagine you rushed up to the child and gave her a push. You would be doing work
on her. Let’s say gave her a push that added 300 J to her energy. We can find out how much mechanical energy she
would have now.
𝑀𝐸𝑖𝑛𝑖𝑡𝑖𝑎𝑙 + 𝑊 = 𝑀𝐸𝑓𝑖𝑛𝑎𝑙
400𝐽 + 300𝐽 = 700𝐽
She now has more mechanical energy. She’s still flat on the ground, so all of this
turns into kinetic energy. She is now moving faster. She has more mechanical
energy. Her potential is the same (she’s on flat ground) so she has more kinetic
energy.
Work can be positive or negative. Positive work adds energy to the object. Negative work takes energy away. Let’s try an
example with negative work.
Negative Work
A car is speeding along a flat road at night. It has 12000 J of kinetic energy. How much mechanical energy does it have?
Yup, 12000 J. Suddenly, a deer jumps out of nowhere. The car hits the brakes, and the friction between the tires and the
road does -8000 J of work on the car. How much mechanical energy does the car have now?
𝑀𝐸𝑖𝑛𝑖𝑡𝑖𝑎𝑙 + 𝑊 = 𝑀𝐸𝑓𝑖𝑛𝑎𝑙
12000𝐽 − 8000𝐽 = 4000𝐽
The car now has less mechanical energy. The potential energy is the same (flat road). Therefore, it has less kinetic
energy. It is moving slower.
No Work
When mechanical energy is conserved, no work is done on the object. The mechanical energy stayed the same (it is
conserved). We practiced identifying these in the last level.
A big hint to identifying these is that the author will usually go out of their way to say there is no friction, no air
resistance (Fair), no thermal energy loss… they are trying to make it obvious that there is no work (positive or negative)
being done on the object.
These don’t actually happen in real life. There is always friction or air resistance or thermal energy. In some situations,
we pretend like there is no work because there is so little we don’t have to worry about it.
Level 3: Using Kinetic and Potential Energy with Work
We can take this one step further. In the last level we learned:
𝑀𝐸𝑖𝑛𝑖𝑡𝑖𝑎𝑙 + 𝑊 = 𝑀𝐸𝑓𝑖𝑛𝑎𝑙
and in a previous section we learned:
𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 = 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑒𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 + 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦
𝑀𝐸 = 𝑃 + 𝐾
If we put this all together we get:
𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 + 𝐼𝑛𝑡𝑖𝑎𝑙 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 + 𝑊𝑜𝑟𝑘 = 𝐹𝑖𝑛𝑎𝑙 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐸𝑛𝑒𝑔𝑦 + 𝐹𝑖𝑛𝑎𝑙 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦
𝑃𝑖 + 𝐾𝑖 + 𝑊 = 𝑃𝑓 + 𝐾𝑓
Example: Finding the Final Kinetic Energy
A car over the top of a hill, where it has a potential energy of 30000 J and a kinetic energy of 20000 J. As it goes down
the hill, the driver hits the gas and the engine does +5000 J of work. At the bottom of the hill, none of the potential
energy is left. What is the car’s new kinetic energy?
What we know:
Pi= 30000 J
Ki= 20000 J
W= +5000 J
Pf = 0 J
Kf =?
𝑃𝑖 + 𝐾𝑖 + 𝑊 = 𝑃𝑓 + 𝐾𝑓
30000𝐽 + 20000𝐽 + 5000𝐽 = 0 + 𝐾𝑓
55000 𝐽 = 𝐾𝑓
Level 4: Real Life
If there is a situation where no outside forces are acting on an object, the mechanical energy is conserved. That means
no friction, nobody pushes on the object, no air resistance, no brakes, no jet packs, no kicks to the rear… nothing. If
nothing tries to speed the object up or slow the object down, we can say mechanical energy is conserved.
In real life, mechanical energy generally does change. Take the example of the swing. If
mechanical energy didn’t change, it would just keep changing from potential to kinetic,
then go back to kinetic. If the mechanical energy didn’t decrease, the swing would go right
back up to the same height it started at. A child on a swing wouldn’t need to pump- they
would just keep going back and forth- rising to the exact same height each time. Remember
the guy in the skate park, how he kept going back up to the same height. The computer was
programmed to have mechanical energy be conserved.
In real life, the mechanical energy of a swing decreases if you don’t pump. Imagine you just sat in the swing. You
wouldn’t go as high each time you went back and forth. Your mechanical energy would be slowly decreasing, until you
finally came to a complete stop. Each time you went back and forth, you’d have a little less mechanical energy, until you
didn’t have any left at all. Friction (thermal energy) is always going to steal a little. It is one of the laws of the universe.
If mechanical energy is conserved, we can use the following equation.
𝐾𝑖 + 𝑃𝑖 = 𝐾𝑓 + 𝑃𝑓
Let’s build off what we learned in our last section. Imagine you lifted a ball with a mass of 4 kg up in the air so that it was
10 m off the ground. Let’s calculate the potential energy below. I’ll put the answer here so that you’ll know if you did it
right or not.
P= 392 J
Now, let’s say you dropped the ball. Just before the ball hit the ground, all the potential energy would be converted into
kinetic energy. So, when the ball hits the ground, it has 392 J of kinetic energy.
Let’s take it one step further. Use this kinetic energy to calculate the velocity of the ball. Remember the mass is 4 kg. I’ll
put the correct answer at the bottom so you can make sure you got it right.
v= 14 m/s