Projectile Motion

I promise, this page is actually pretty interesting! Projectile Motion, governed by parabolas and elipses, is actually a really cool topic that I'm sure everyone will enjoy!

A Short Introduction to Gravity

Before we learn about falling objects, we need to learn about what makes them fall - gravity. Gravity is an attraction force that pulls objects together. Earth is round for this reason - it has pulled itself together from all sides as far as it can go. In fact, planets also have effects on each other's orbits - known as perturbations - when they pass by each other. Their gravitational fields attract each other, which changes their overall orbit around the Sun.

The Law of Universal Gravitation is a mathematical equation that allows you to discover the amount of gravitational attraction between two objects. It goes like this:

F = G(m1 x m2)/d^2

Where F is the force between the objects, m1 is the mass of the first object, m2 is the mass of the second object, d is the distance between the two objects, and G is the gravitational constant. G is amount of gravitational force between two 1 kilogram objects 1 meter apart. More accurately:

 G = 6.67 x 10^-11 N*m^2/kg^2

Not only is the number insane, the unit of measurement is too 💀

The Inverse Square Law

The relationship between gravity and distance is the inverse square law, which states: 

If the distance between two objects increases by a certain factor, the gravitational force decreases by the square of that factor. 

This law applies to any force that spreads uniformly outward from a single source into the area surrounding it. In other words, the weight of a rocket depends on how close it is to the surface of a planet - the further it is, the smaller the effect of gravity. However, its mass always stays constant. Weight is really just the force an object exerts on a supporting force, so when you are in free fall, you feel weightless since you are not exerting a force on a supporting force (the earth).

Projectiles

Projectiles have curved paths because of gravity. If you throw a projectile without gravity, it will simply fly in a straight line forever and ever until something disturbs it.

When gravity does exist, projectile motion is altered. If a projectile is thrown straight up, it falls back to its original spot instead of flying up forever in a straight line. If a projectile is thrown horizontally, it is both propelled forward by its own inertia and pulled down by gravity, creating a curve known as a parabola. For now, we are neglecting air resistance.

If two projectiles are dropped, they will hit the floor at the same time if their vertical distance from the ground is the same, Regardless of whether they were thrown horizontally or dropped vertically, as shown in the image below.

Let us continue. When projectiles are launched at angles, their paths trace out parabolas. When you throw a projectile at a constant force, the angle you shoot it at determines how far it will land from you. In fact, if you throw it at a 60 degree angle, it will land at the same place as it would if you throw it at a 30 degree angle. Notice that they sum to 90 degrees. No matter which angle you throw it at, it will land in the same location as it would if you threw it as its complementary angle. Obviously the altitudes reaches and the time in air is different. The best launching angle is 45 degrees, assuming air resistance is negligible.

 When air resistance is not negligible (welcome to the real world, babe), objects take smaller, shorter parabolic paths, shown in the mortifying picture below (I'm sorry guys).

When projectiles are heavy, however, the launching speed at which you fling them will be different - if you hit it at a 60 degree angle, you will have a slower launching speed than if you throw it at a 30 degree angle. For this reason, the farthest throws are typically 45 degrees and below. Cool! Now we can go yell at those pro javelin throwers in the Olympics to make yourselves feel better.

When you throw a projectile, the time it takes to go up is the same as the time it takes to go back down, not counting air resistance.

Satellites and Orbits

Now, if you're a nerd like I am, you probably suck at flinging footballs around. (Hence the reason I have to depend on my runs to bring my PE grade up ToT).

If a ball is flung by me (weak), it will land somewhere in front of me. If I throw it with more force, it will land farther away. Common sense.

But what if I throw it with so much force that it goes around the entire Earth? What if I throw it fast enough that it keeps going around the Earth, stuck in this eternal dance between speed and gravity? 

Well, my dear friend, if I somehow manage that with my nonexistant muscles, the object would then turn into a satellite. Satellites are simply objects that are stuck in this dance - falling continuously around the earth, too fast to fall into it but too slow to escape orbit - just like the Moon.

A satellite always moves perpendicular to the center of gravity it is orbiting. In order for a projectile to begin orbiting, it must be traveling at 8 km/s and be traveling perpendicular to its center of gravity. Obviously this is neglecting air resistance. Every projectile thrown on Earth has to deal with air resistance making its life difficult. Even the ISS, and that was made by total nerds (I say that with the most respect and admiration. You guys are the real ones).

Fun fact: The ISS is launched far up to escape most air resistance, but there is a little atmosphere that scrapes against it. Because of this, it must be boosted back into orbit every once in a while.

Elliptical Orbits

When an object on Earth travels faster an 8 km/s, but slower than escape speed, it orbits in an ellipse. In an ellipse, the sum of the distance from two points (foci) must be constant. The closer the foci, the more circular the orbit, and the further the foci, the more ellipitcal the orbit.

Basically, if an object is traveling at a speed where it is still capable of orbiting, it is flung far away from its planet or star before returning, thus creating the elliptical orbit we are talking about. Please look to the picture below if you are confused.

Escape Speed

Escape speed or escape velocity is the speed that enables an object to escape Earth. It is 11.2 km/s. If you throw something at speeds greater than 11.2 km/s, you will never see it again. Sorry man. Should've told you earlier.

The escape speed for the solar system is 42.5 km/s. if you toss an object at a speed between 11.2 and 42.5, it will leave Earth orbit and begin to orbit the Sun. But if your speed is larger than 42.5 km/s, say goodbye, because you will certainly not see that thing again. 

These escape speeds are initial speeds. Given enough time, an object can leave solar system orbit without reaching these speeds. However, these speeds are convenient because they cut down travel time. Just like me on my bike.

Either way, it was wonderful to be able to teach you all. I hope you learned something new! See you in my next page!