Welcome back to the grind, PaleoPosse! Did you have a good Thanksgiving vacation? I sure hope so, because today I’m going to flex your brain muscle and attempt to teach you some REAL science: PHYSICS. Not that mushy, gushy “biology” and “paleontology” pseudo-science that Ryan and Patrick peddle.
AND if you stick through it all the way to the end, there’s a prize in store for somebody!!!
But enough of that, today I’m going to teach you about the Work-Energy Theorem. Otherwise known as, “Everything they tried to teach you in high school physics.”
Work & Energy – What are they?
In order to understand the Work-Energy theorem, you need to understand the separate definitions of both Work and Energy.
Work = Force x Distance
Taken in context, Work is an action that is performed on an object or system. So a continuous force applied to an object over a given distance constitutes “Work” performed on that object.
To illustrate, I will call upon my friends Dink and Donk. Dink and Donk don’t really care about physics, but they like to have fun. So D&D decide to go do some Jackass-style shopping cart races (or shopping carriage races, if you’re from the NorthEast) at the local supermarket. Dink is in the shopping cart and Donk pushing him.
Let’s assume that Donk can push Dink and the cart with 10 lbs of force. He pushes for 40 feet, because that’s how much parking lot there is until the cart hits the curb.
If Donk maintains 10 lbs of force over the entire 40 feet, he will have performed 400 ft-lbs of Work.
10 lbs x 40 feet = 400 ft-lbs of Work
(or 542.3 Joules for those of us who like to work in SI units)
So when Dink and Donk get home and their father berates them for jerkin around, and tells them to get a job, Donk can legitimately say, “But Dad! I did a lot of Work out there! 400 ft-lbs of it!”
So now that you understand Work, what is Energy?
Energy = 1/2 x Mass x Velocity²
*Technically this is just “Kinetic” energy. In reality, Energy can be expressed in kinetic energy, potential energy, thermal energy, pressure changes, or chemical energy. But for the purposes of today’s lecture, we’re going to focus on just Kinetic energy.
So taken in this context, Kinetic Energy is a characteristic which an object or system has at a given point in time(as opposed to Work, which is an action performed on an object or system). So if you know an object’s mass and it’s velocity, then you can accurately calculate that object’s current Kinetic Energy.
In the case of Dink and Donk, we can can determine their Kinetic Energy at any point along their 40 foot journey.
For the sake of simplicity, let’s say Dink and Donk both weigh 150 lbs (4.662 slugs, the English unit for Mass), and the shopping cart weighs 20 lbs (0.622 slugs). Let’s also assume that Donk is a pretty fast runner, and can get this puppy up to around 7 miles per hour (or around 10 feet per second). So what is their combined Kinetic Energy?
1/2 x (9.946 slug) x (10 ft/s)² = 497.3 ft-lbs of Energy
(trust me, the units work out…stupid english units)
Now there’s something interesting here that I hope you’ve noticed: the units of measurement for Work and Energy are THE SAME. Ft-lbs in our case, or Joules in SI units. This is the essence of the Work-Energy Theorem.
The Work-Energy Theorem – Equating Work and Energy
What we have shown in the examples above is that Energy and Work are two completely different concepts, yet they are expressed in the same units. What does this mean?
It means that Work and Energy are two sides of the same coin. Work creates Energy, and Energy performs Work.
Therefore we can equate the two formulas
Work = Energy
Force x Distance = 1/2 x Mass x Velocity²
This allows us to perform some pretty incredible calculations.
In the case of Dink and Donk, we can calculate how far Donk will need to push Dink in order to reach a speed of 10 feet per second. All we have to do is re-arrange the formula to solve for Distance.
Distance = [1/2 x Mass x Velocity²] / Force
Distance = [1/2 x (9.946 slug) x (10 ft/s)²] / 10 lbs
Distance = 49.73 feet
Pretty cool huh? Well shit’s about to get even cooler…
Using some incredible Math-Magic, we can transform the Work-Energy equation into a set of other equations called the “Kinematic Equations”.
Using the Kinematic equations, we can predict a plethora of different kinetic characteristics.
For instance, in the case of Dink and Donk, we can predict how far Dink will fly after hitting the curb (assuming there is a 6 foot drop after the curb, and Dink is currently 1 foot off of the ground).
Now the following math steps are generally considered to be the most annoying calculations that a high school Physics student will ever have to do. (Who REALLY wants to use the quadratic formula? BLEGH!) But if you’re interested in Physics or Engineering, don’t worry! Once you add Calculus into the mix, these equations get A LOT simpler and easier to handle. But for now, I’ll solve them for you!
So first we have to think about the problem. We know that Dink’s initial velocity before striking the curb is 10 ft/s. Assuming Dink does not experience air resistance, his forward velocity WILL NOT CHANGE as he falls. In other words, his final velocity is also 10 ft/s. In addition, he is not being pushed while he is in the air, so his lateral acceleration is 0 ft/s².
Vi = 10 ft/s Vf = 10 ft/s a = 0 ft/s²
So let’s look back at our equations now… we have Initial Velocity, Final Velocity, and Acceleration – and we’re trying to solve for distance… all we need now is Time.
How long does Dink stay in the air? That is determined by gravity and the height that he falls. Since we know that gravity accelerates downward at 32 ft/s², and that Dink has no initial vertical velocity, we can solve for the time it takes Dink to fall using the upper left equation shown in the Kinematic equations.
So now we have our falling time, and we can now solve for our lateral distance, d!
Work-Energy Theorem – COMPLETE!
So look back at what we’ve done here.
- We calculated the Work performed by Donk while pushing Dink and the shopping cart.
- We calculated the Kinetic Energy of the Dink-Donk-Cart assembly.
- We calculated how hard Donk would have to push to get Dink and shopping cart up to 10 ft/s.
- We calculated how long it will take Dink to fall from the 7 foot ledge.
- We calculated where Dink will land after he’s ejected from the shopping cart.
Now I don’t know about you, but I’ve had entire homework assignments with less calculations than this. This is a feat to be proud of.
But more importantly, I hope you’ve taken away a ground-level understanding of the Work-Energy Theorem. If you know how much energy you want a system or object to have, then you can calculate how much work you need to perform to get it there. Conversely, if you know how much energy a system has, you can calculate how much work that energy can perform. This has literally MILLIONS of real-life examples:
- The energy of wind movement performs work when it turns a Wind Turbine
- The chemical energy in gasoline performs work on a piston, which in turn performs work on a vehicle to create kinetic energy.
- Work is performed on air as it enters a Jet Engine to speed up the air, which results in higher kinetic energy of the air particles, which pushes the airplane
- Stirring a pot of water performs work in the form of heat transfer, which results in a higher temperature in the water (higher temp = higher energy)
- When you throw a water balloon at someone’s face, their face performs work on the balloon, which then increases in pressure (higher pressure = higher energy) until it pops.
So here’s the skinny PaleoPosse. I want to make a video demonstrating this principle. If you can think of some really good examples of the Work-Energy theorem that I could do myself and record (without having to buy anything TOO expensive), I will pick a favorite and send them a prize! (I do, however, reserve the right to pick NONE of your ideas, if I happen to have a really good one.
I hope you’ve enjoyed this week’s blog post! I’ll try to do these “Real Science” posts more often now.