Energy and Momentum

Alright guys! Now we're onto our last topic in Mechanics! After this, we get to more fun stuff, like E&M and Fluid Mechanics. But of course, we have to cover this as well.

Momentum and Impulse

Momentum, simply put, is the amount of inertia of a moving object. Momentum is the mass of object multiplied by its velocity, or:

Momentum = mv

Thus, lighter objects can have more momentum than heavier objects if they are moving faster. Note that if a ball lays on the floor, unmoving, it has no momentum at all. Applying a force for long amounts of time increases the momentum. Basically, when you throw a ball (I know, how original of me), and you hold the ball until the last second, it will have more force than if you just fling it with your wrist.

Impulse is the change in momentum of an object. It is the amount of force on that object multiplied by the time in which it is enacted. In other words:

Impulse = Ft

or Ft = ∆mv

In which ∆ means "change in." When an object's impulse is high, its momentum is too. When a ball is thrown with a lot of force and it hits a wall, the time interval in which the momentum changes is very small, resulting in a large force hitting the wall. Newton's Third Law tells us that this force hits the wall and the ball equally, which is the reason it bounces off. Similarly, when you throw a ball and it hits soft foam, the time interval in which the impact is made is relatively large, thus decreasing the force on the object.

Another prime example of this is with car airbags. Airbags extend the time interval in which the large force of the crashing car acts, thus reducing the force significantly and keeping you alive. In the same way, when the airbags do not come out, the time interval in which your body hits the front of the car is very short, which increases the force on your body.

But it's not over yet! Now you're in trouble. The car you crashed just bounced. Bouncing increases impulses significantly. It takes more impulse for something you hit you and then bounce off than it does for it to just hit you. Well, I hope you're okay - I need viewers for this website after all 😈

Conservation of Momentum

The law of conservation of momentum, according to Conceptual Physical Science by Paul G Hewitt, John Suchoki, and Leslie A Hewitt (cited on the welcome page), is "in the absence of an external force, the momentum of a system remains unchanged."

This means that the total momentum of a system is constant, as long as an external force does not get involved. Momentum is also a vector quantity, since it utilizes both magnitude and direction.

As we now know, net momentum is always constant. Similarly, impulses are equal and opposite (the ball hitting the wall example mentioned earlier showcases this). Everything follows Newton's Laws of Motions! Use them to your advantage just like I did when I was drinking coffee in the car. I never had any coffee while the car was slowing down (just in case it came splashing back once the car came to a full stop). Instead, I enjoyed it while the car was accelerating (carefully) and the coffee automatically spilled into my mouth. I didn't want to stain the white carseat we were working so hard to preserve. Now you know how useful physics is in real life.

Potential and Kinetic Energy

When two objects collide, because of the conservation of momentum, they either both move or one stops and the other moves with the same momentum. Elastic collisions are ideal collisions - the colliding objects experience no net loss of kinetic energy while hitting each other. However, reality is often not what we want it to be, and collisions are inelastic. The colliding objects do experience a net loss of kinetic energy, in heat or deformation. The best version of this collision would be if the two objects stuck together instead of going anywhere.

However, before I explain what kinetic and potential energy really are, we need to discuss work and power. Let's go over some vocabulary.

Work is the change in kinetic energy. In a mathematical sense, work = force x distance, W=Fd. An example of work is shown when a family is moving boxes. Pushing, lifting, and pulling a box are all forms of work. However, holding a box over your head is not doing work to it. Lifting it up is doing work, but just holding it above your head is not.

The work-energy theorem is a theorem that states that work is the change in kinetic energy.

Joules are the unit of work. A joule is the work done when 1 N of force is used to move an object 1 meter.

Power is the amount of work done in a time interval. So, power = work done / time interval. Doing work fast takes more power.

Power is also the rate that energy switches forms. 

The unit of power is a Watt. 1 watt is equal to 1 joule per second.

Now that we've gone over the background information, let's actually discuss kinetic and potential energy. Potential energy (PE) is stored energy. Fuel has potential energy, a ball on a table has potential energy, and a stretched slingshot has potential energy. 

Gravitational Potential Energy (GPE) is potential energy due to the elevation. The amount of PE an object has can be calculated with the following equation, where mg represents the weight, and h represents the height from the ground:

PE = mgh

Coming back to the concept of work, when work is done, energy is exchanged. When we do work to that box from earlier, we change PE (stored energy) to Kinetic Energy (the energy of motion). When an object has kinetic energy (KE), it is in motion. If that ball on the table rolls and starts to fall towards the ground, its PE is converting to KE. To calculate the amount of KE an object has, you can use the equation:

KE = 1/2mv^2

Where m is the mass and v is the velocity of an object. However, KE is also the net force an object has multiplied by the distance. Substituting KE for Fd (force x distance), we have:

Fd = 1/2mv^2

Now, how are kinetic and potential energy related? When an object, like the falling ball we talked about earlier, go from a state of rest to a state of motion, potential energy turns into kinetic energy. When the ball is nearing the floor, its potential energy is the lowest and its kinetic energy is the highest. In a swinging pendelum, the kinetic energy is highest while the pendelum is at the bottom of its cycle. When it swings down, PE converts into KE, and when it swings back up, KE converts into PE.

The total KE and PE added together is always constant, except that friction turns KE into heat. For example, if something is sliding along a carpet (KE), it eventually comes to a stop because of the KE turning into heat energy due to friction. It is the reason why your socks feel hot when u slide on a carpet.

However, sliding on a carpet is not energy loss. It is simply energy changing forms, which brings us to our next topic.

Conservation of Energy

The law of conservation of energy tells us that energy cannot be created or destroyed. It can only change forms. In the carpet example from earlier, the kinetic energy changes forms, turning into thermal energy, but it is not lost. Any form of energy can be converted from one form to another.

Also, please remember that momentum and KE are different. Momentum is a vector quantity and can be canceled out. Energy cannot. 

Basically, energy always has a safe Twitter life but momentum might be using a burner account to hide from public attention.

Momentum depends on velocity, KE depends on velocity^2, Energy changes forms, but momentum does not. This all goes to say that they aren't the same thing. If that's confusing, don't worry. I'm confused too. My friends say that my default facial expression is confused. We're in the same boat.

Simple Machines (a short bonus section)

Machines multiply or change directions of forces. They do not create energy. They simply effectively take advantage of work, force, and distance. The lever, for instance, increases distance to decrease force required (like a seesaw that was put on wrong). Take a look at the picture below to better understand.

Basically, the lever takes advantage of this rule:

Applied force x Applied distance = Output force x Output distance

If you apply a little bit of force, but increase the distance you exert it over, you can decrease the output distance, thus increasing the output force. In other words,

Applied force x Applied distance = Output force x Output distance

Big brain 🤯

Now, my last concept (before you can finally move on to E&M, Fluid Mechanics, and Thermodynamics, and Waves. I love waves) is efficiency.

Efficiency = useful energy output/total energy input

Basically, efficiency is the ability to tell if you're wasteful or not. I'm pretty sure you're not efficient. I bet you litter in national parks. How dare you. 

It's okay. It's impossible to be completely efficient anyway. Energy is wasted as heat due to friction, so having a machine with 100% efficiency is not feasable.

But still. Don't litter.