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.

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**Linear motion** is the motion that is natural to an object: moving in a straight line. According to Newton s First Law of Motion, an object not affected by any force will continue indefinitely in a straight line. If a projectile is thrown vertically, it will travel in **linear motion** and will begin to fall when the force of gravity equals the force of the throw.

Let's talk about linear motion, here's an example of linear motion I take this tennis ball and I throw it up and down it's moving in one dimension okay. That's a little different than parabolic motion where it's moving at an angle and the force of gravity is also pulling it down. So let's first look at linear motion, again simply put it's just change is distance over the change in time okay.

A problem you might see related to linear motion is a ball thrown straight up at 20 meters per second for how long will that ball move up and how high will it go? How far will it travel? Okay to solve that we need to think about it so, we're throwing the ball up at 20 meters per second and it's going to up until the velocity due to gravity is also 20 meters per second. So now it's at the point at which its velocity due to gravity has equaled to velocity due to the initial velocity applied to the ball.

Okay, well again to solve that let's look at a couple of equations okay first off the distance traveled due to a simple velocity is just a velocity times time, so meters per seconds over seconds okay. The difference due to gravity is 1 half gravity times time squared, that'll be useful for the second part here okay. And again the force of gravity is 9.8 meters per second squared. And to solve problems like this it's useful to simplify, so let's simplify that to 10 meters per second squared. That'll be close enough for our calculations here okay so I want to know at what point will my initial velocity equal the velocity due to gravity. So my initial velocity is 20 meters per second and I want to know at what point will that equal 10 meters per second squared times t okay.

Pretty easy problem, if I see the 20 meters per second equals 10 meters per second times t I can see that seconds squared and this is going to be unit in seconds so they'll cancel. I can see that my t equals 2 seconds okay. So my ball is going to stay in the air for about 2 seconds okay, before it starts to come done. Now if I'm asked how long will it remain in the air total, well that's going to be 2 seconds up and 2 seconds down okay. So that answers part one, now let's look at number two.

How high will it go in the air okay, well now I want to figure out the distances and again distances for velocity is velocity times time, distances due to the force of gravity is one half gravity times time squared okay. So how do those two distances because my velocity going up is going to start at 20 my velocity due to gravity is going to start at zero and then it's going to slowly increase, it's going to continue to increase. So I'm going to add those two vector, velocities together okay so if I say and for that one remember the gravitational velocity is going to be a negative value whereas my initial velocity up is going to be a positive value. So let's go ahead and calculate that velocity times time is 20 meters per second times 2 seconds okay and minus gravity pointed down and again the distance due to gravity is one half and 10 times 10 meters squared that's the force of gravity times time squared okay which is 2 times 2.

Again simplifying further, this is 40 meters per second times 2 seconds okay so the seconds are going to cancel there minus 1 half times 10 times 2 squared okay and again to simplify that 10 times 2 is 20 and one half of that is, I'm sorry 2 squared is 4 so this is 40 and 1 half of that is 20, so I've got 20 meters okay so my answer here is the ball goes 20 meters in the air before it starts to descend.