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Parabolic Motion

Teacher/Instructor Matt Jones
Matt Jones

M.Ed., George Washington University
Dept. chair at a high school

Matt is currently the department chair at a high school in San Francisco. In his spare time, Matt enjoys spending time outdoors with his wife and two kids.

If an object moving forward in a straight line is affected by gravity it will fall in a parabolic arc. Since projectiles are objects affected only by gravity, the path of a projectile moving forward from the momentum of an initial thrust is parabolic. When working with parabolic motion, some important equations to know include change in x = the horizontal component of the vector X time and the vertical component of the vector = gravity x time.

Projectile motion here's a ball I fire it into the air, boom and it travels in a parabola hence the name parabolic motion right. so it's basically any object that's fired into the air and when it moves there's a lot of different forces acting on it. It has got a vertical velocity, it has got a horizontal velocity. It's got of course gravity pointed down so when we solve problems with parabolic motion we need to do some things we need to figure out what that horizontal and vertical velocities are before we can move on and solve a problem. So let's look in an example, let's say I've got an object maybe I'm firing it out of a cannon and I'm firing it at 50 meters per second at 37 degrees above the horizon okay?

Well what I want to figure out is what's my horizontal velocity which goes along the x axis and what's my vertical velocity which goes along the y axis. Okay if I know that it's 50 meters per second at that angle and that angle is 37 degrees. I can use some trig to solve that okay so if I want to solve for x okay I know that x over this value is basically the cosine, the cosine of 37 so if I multiply this value by the cosine of 37 okay I have 50 meters per second times 0.08 that's going to give me 40 meters per second. That is my velocity in the horizontal okay that's how long it's fastest moving along the x axis okay.

Now I want to figure out what the y velocity is, the vertical velocity okay and again for that one I can use the sine of 37 because this over this equals the sine of 37 degrees okay. So to solve for that I've 50 meters per second times the sine of 37 degrees which is 0.6 and that's going to equal 30 meters per second okay. So now I've got I'm going to go ahead and write them right here this guy equals 30 meters per second, this guy equals 40 meters per second. Now I've got some good information for solving a parabolic motion problem.

Okay, so let's look at an example, I've got my object, my canon ball I'm firing at 50 meters per second at a 37 degree angle above the horizon, how long before it reaches its maximum height. Okay, well in that case I'm looking at those velocities that are causing it to go up which is my y velocity 30 meters per second and a velocity which is going to cause it to go back down and I know it's going to stop moving up when that velocity 30 meters per second equals the velocity due to gravity. Okay, at that point it'll stop moving up and it'll start moving back down. So I know that my y velocity is 30 meters per second okay. And I know the force of gravity in this case I'm going to simplify I'm just going to say is 10 meters per second squared, don't forget to square it times t okay, now and t is in unit in seconds.

Okay, and when I multiply that out my seconds here I'm going to cancel with one of those so I just have meters per second okay, and I have 30 meters per second equals 10 meters per second times t and that's a pretty easy problem t equals 3 okay. Good so that means it's going to spend 3 seconds going up before it starts coming down okay. Now how high does the object rise okay so again to calculate the distance it's moving up I have 2 velocities that I want to calculate right, the first one is the velocity going up, vertical velocity is 30 meters per second and the second one is t okay I'm sorry times t and then the second velocity is one half of gravity. The force of gravity which is going to be pulling it down.

So to solve that I say 30 meters per second and my time is 3 seconds times 3 seconds okay and I'm going to subtract the velocity due to gravity because that's pulling it in the opposite direction right, which is one half 10 meters per second squared and times t squared so this is going to be 3 seconds squared which is going to equal 9 seconds squared okay. My seconds squared cancels with that okay and 10 times 9 is 90 and one half of that is 45 okay my velocity going up is 3 times 3 cancel my seconds so I get 45 meters is pointed down and 90, 3 times 3 is 90 going up so my answer is 45 meters. So my can ball rises 45 meters in the air.

Okay, alright well now I know a lot of information and now let's look at a couple of more problems that you'll probably going to get asked okay? Another question might be well how long is that can ball going to spend in the air before it lands? Okay, well to figure that I've got the information if it's going to spend 3 seconds going up okay 3 seconds going up it's going to spend the exact same amount of time coming down. Okay, so my answer there is 6 seconds okay. Alright now another question that's commonly asked is how far is that going to land from where I shot it right I want to know where my canon ball is going to land maybe hopefully on my target.

Okay, so that distance can be calculated by the velocity on the horizontal okay which is right there okay times time. So if I calculate that, I've got 40 meters per second times 6 seconds. My seconds cancel and I get 240 meters my canon ball has fired 240 meters away okay. And that's how you solve a problem and many steps of a problem using parabolic motion.