The three types of motion

Dropped
Thrown
In two directions
Example
Real Life

If there is gravity, when an object is released, gravity will pull the object downward. On Earth gravity is -9.81m/s/s. Traditional experiments use this for G, thus eliminating one variable. If you were on a different planet, however, then you would need to adjust G accordingly.  The effects of G have fascinated scientists for years. Everyone knows about Newton and the apple that fell on his head and knocked him out because he assumed G was higher than it was. They also know about Galileo, who dropped stuff from the leaning tower of Pisa. They say he dropped important stuff, but he probably dropped water balloons too. The first Physicist, however, is Aristotle. For our purposes, we must assume air resistance is negligble. We must assume we are acting in a vacuum or a place with very non-resistant air molecules. If we are to remain on Earth, we must assume that the Spaceballs have succeeded in their plan to steal all the air from Druidia, but they went to the wrong planet, and they sucked all of the air out of the Earth.
 

Dropped Objects

A dropped object's distance and velocity can be found using the following formula.

D=½G*T²+D0 where D0 represents the initial vertical distance away from the origin, and T represents the time since the begining of the measurement. The Velocity can be found using the equation. Modifying D0 does not have an effect on the velocity and will only move the graph up and down. There is a relatively complex reason why this works. It is called derivatives.
V=G*T
Distance vs. Time                 Velocity Vs. Time
    
Graphs where D0 changes from 4 to 6 to 8 to 10.

To find either the time it takes to fall a certain distance or the distance it falls in a certain time, simply plug in the known variable and solve for the unknown one. It is important to realize that Distance Vs. Time graphs are graphs of parabolas. When there is no initial velocity, the parabola's vertex is located on the Y-Axis, at time 0. All the negative times will have the same distance from the origin as their respective positive values. This is a parabola with respect to time, only. When it is looked at with respect to the horizontal position, then it is a straight line down.
 

Thrown Objects

When an object is thrown, it has an initial velocity other than zero. If it is thrown straight up or straight down, then it can follow the formula

D=½G*T²+V0*T+D0, where G, T, and D0 represent the same things they did in the dropped objects. V0 represents the initial velocity. If the initial velocity is negative, then the object was thrown downward. If the initial velocity is positive, then the object was thrown upwards. The Velocity can be found with the equation

V=GT+V0

Distance Vs. Time                 Velocity Vs. Time
    

With a positive initial velocity, the parabola created is moved to the right, and the maximum height can be seen.
 

Notice that the Velocity vs. Time graph has the same slope with an initial velocity as it does without.

Distance Vs. Time                Velocity Vs. Time
    

With a negative initial velocity, the parabola created is moved to the left, so the vertex becomes invisible within the window. The slope of the Velocity vs. Time graph is yet again the same, but now it starts below the X-Axis.
 

Motion in Two Dimensions

Sometimes, objects do not fall straight down.  They are given a small push horizontally and fall parabollically, rather than in a straight line. In this case, it makes a parabola in reality, or on the graph of Y-position Vs. X-position. To find the parabola, we must use the parametric mode of the calculator. Parametrics creates a Y postition vs. X position graph with time as a variable that is used in two separate equations. To find the Distance that an object has travelled in both directions, you must use the two following formulas together

X=V0*Cosø*T+X0
Y=½G*T²+V0*Sinø*T+Y0

V0 is the initial velocity of given equation. If you know the initial X and Y velocities, V0 is found be V0²=Yv0²+Xv0²
ø Theta is the initial angle of the motion. If you know the initial X and Y velocities, ø is the Arctan(Y/X)
X0 is the initial X position with respect to the origin. This does not affect the appearance of the parabola, only the location.
Y0 is the initial Y position with respect to the origin. This does not affect the appearance of the parabola, only the location.
G is our good old friend
T is the independent variable that is used to create said parabola.

X-position Vs. Y-position with       List of X's and Y's
respect to time
    
 

Example Problem

Let us take, for example, the problem of holding a glass in your hand while standing over a hardwood floor on this Spaceball ridden planet Earth. This is a glass of let's say beer and it is definitely not your first one. You stagger around, when suddenly, due to ineffective muscular control and the slipperiness of the glass, it falls from your hand that is two meters high. Your sharp physics mind springs into action. The glass will break if it hits the ground going more than 5.9 meters per second. At what time will it reach the ground and at what veloicty would it be going? Will the glass break?

First, we must determine an attack pattern. I recommend working backwards. In this scenario, we want to know if the glass will break. So we need to know if V at time T is greater than or less than the absolute value of (-5.9). To find that we need to know V at time T, but we don't know T, we need to know T when D=0.

D=½G*T²+D0
0=-4.91T²+2
2/4.905=T²
T=.639sec

V=GT
V=-9.81*.639
V=-6.269m/s

Absolute value of -6.269m/s>Absolute value of -5.9m/s, Therefore the glass would shatter.
 

Application to real life

In the real world, these concepts have enormous application. They can be used to check the velocities of falling objects, such as Michael Jackson's baby, or they can be used to plot the path and final location of objects dropped from say airplanes. It can test a variety of scenarios, and the use of formulas allows for quick adjustments to be made. You can also check to see how a bullet might travel when shot on the moon, or how fast a baseball is going as it hits the bat, then how fast it's going when it goes over the fence.

Of course, these are not all of the aspects of motion. There are many others. Some are directly realted to air resistance, and some are just very strange to hear about.

I am Samir Jenkins. I am a student at Fayetteville High School who made this page all by himself because it was required for my Physics class. Please, feel free to e-mail my teacher David A. Young at DYoung@fayar.net. Although Mr. Young doesn't know everything about Physics (but don't tell him that), he does know a lot. He even has his own physics webpage. There are other websites, however, that could supplement his knowledge.

 Also, in case you don't appreciate my subtle use of links, here they are again.

Newton
Galileo
Aristotle
The effects of negligble air resistance
Spaceballs
Derivatives
When a bullet is shot on the moon
Tumbling and fluttering motion
Young's Physics site
Top 20 Physics