First, I’ve never taken a physics class or attended highschool before, hence the ELI5. I’ve read many explanations but it doesn’t really make intuitive sense to me. For example (assuming there’s no air resistance / drag), let’s say I was traveling in a car going 120 mph and I wanted to decelerate to 90 mph. This would take four times as much energy than going from 30 mph to 0.
But let’s say there were two cars traveling at 120 mph. The car next to me decelerates to 90 mph, but I’m still going 120. From my point of view, the car next to me just started going 30 mph in the opposite direction. Why would this require 4 times as much energy than if both cars were just stationary, and the car next to me actually started going 30 mph in the opposite direction?
And, let’s say we’re both standing on earth. One person at the north pole and one at the equator. Both of us throw a ball, but the ball at the equator is already traveling at something like 1,000 mph due to the earth’s rotation. Shouldn’t throwing a ball eastward then require way more energy to go from 1,000 to say 1,020 mph, than the person throwing the ball at the north pole who just has to accelerate it from 0 mph to 20 mph?
In: 3
I also had this problem. It seems unintuitive that somehow going from 50 to 60 mph adds a different amount of oomph to an object than going from 60 to 70 mph.
I’m going to base my answer on [this](https://physics.stackexchange.com/a/14752) physics stackexchange answer. I’ll rephrase it to make the mathematical steps more clear, but I recommend reading that answer as well if you know high school algebra well.
Hopefully you have the intuition that energy is some property that objects have which can cause internal transformations in matter. Whenever an object collides and as a result deforms or heats up, that’s kinetic energy acting on the object.
So let’s look at a situation that makes it easy to account for all the kinetic energy – a clay ball hits a wall. It will splatter, and the size of this deformation can be used to calculate the energy. If two clay balls of equal size hit a wall and splatter to the same extent, they had the same amount of energy. Let’s say the balls were moving with speed **v**, and had an amount of energy **E**.
Our first step is to change this experiment. Instead of throwing the balls at the wall, we throw them at each other. They splatter in the air instead of on the wall, coming to a complete stop. They were both still moving with speed **v**, but in opposite directions. When we do this experiment, we find that the amount of deformation each ball undergoes in this collision is the same as when they hit a wall. This makes sense – each ball has energy **E**, so the entire collision has energy **2E**, but it’s spread out through twice as much mass. Each ball undergoes an **E**’s worth of explosion, the same as if they each hit a wall.
Our second step is to change this experiment once again. Throw the balls a each other, but now observe them from a car moving forward at speed **v**. The ball we’re following looks like it’s hanging in the air, while the other ball is moving toward it at speed **2v**. Now to see what happens when they collide, you’ll need to keep track carefully. To a person standing on the ground, the two balls stopped moving completely, and fell straight down. But since we’re in a car moving forward at speed **v**, the collided balls are now moving backward at speed **v**. From the car, it looks like the **2v** ball came in, collided, and knocked the combined system backwards.
Now the crucial step – other than this change in relative speeds, the collision looks identical. Us being in the car doesn’t change the events. The splatter of each ball is the same size. So the moving ball went from **2v** to **v**, and yet it delivered **2E**’s worth of energy. Going from **0** to **v** grants an object **E** worth of energy, and yet going from **v** to **2v** grants *at least* **2E** (in reality more, since the two-ball system is still moving, so it didn’t lose all its kinetic energy to the splatter). We got here not by assuming any kind of mechanics, but by assuming that the laws of physics work the same when we’re moving as when we’re stationary – if we do all our math from a moving car, we should still observe the same things happening, even if we observe them happening at different speeds.
To get the exact quadratic relationship, we have to account for the fact that the two balls are now moving backwards with velocity **v**, which means there is an extra **2E**’s worth of energy stored in their motions. That’s a total of **4E**. If velocity **v** has energy **E**, while velocity **2v** has energy **4E**, that means doubling the velocity quadruples the energy, which is only possible if **E(v) ~ v^(2)**. Then **E(2v) ~ (2v)^2 ~ 4 v^2**.
This argument suggests that this relationship does not come from the inherent properties of objects, the way electromagnetic energy does. That energy is somehow stored in the bonds between nuclei and electrons. Kinetic energy is some kind of property of the geometry of space and time, so just thinking about the symmetries of motion can tell us what the relationship between motion and energy must be.
Latest Answers