Gravity and Quadratic Equations

Intro
When we finished talking about Gravity last time, the astronauts on Apollo 11 confirmed Galileo's theory that Gravity causes all objects to accelerate at the same rate (regardless of weight or size).

Near the surface of the Earth, the acceleration of an object due to gravity is: 32 ft/s² (9.8 m/s²). You would read that as "Thirty-two feet per second squared" (or "Nine point eight meters per second squared"). But you might be asking: What is a second squared?

[Note: Remember that as you move farther and farther away from the Earth, it has less of a gravitational pull on you. That change is only very slight even when you are as far from the Earth as in an airplane. All the way out near the moon though...that's a different story.]

Another way to write 32 ft/s² is 32 ft/s/s, which you would read "Thirty-two feet per second, per second." We know that feet per second is just a regular old speed. A car driving 50 miles per hour (mph) can also be said to be going 34 ft/s; every second, the car moves forward 34 feet, and every hour, the car drives 50 miles.

Feet per second is how far something moves after one second. So, feet per second, per second is how much faster something moves after one second. An acceleration, then, is the change in speed. If I am holding a bowling ball (over the edge of a cliff) and let it drop toward the Earth -- which has an acceleration due to gravity of 32 ft/s² -- then the ball starts with a speed of 0 ft/s (at the moment I let go of it) and after 1 second of falling, the ball is going 32 ft/s.


So finally, we have reached our Quadratic Equation. The distance equation is:
X_final = X_initial + V_initial*t + (1/2)*A*t²Where X_final is the final location of an object, X_initial is its initial (starting) location, V_initial is the initial (starting) velocity, A is acceleration, and t is time.

Physical Meaning of the Equation
You may remember the old distance equation that you learned in Algebra I: D=R*T (ie Distance equals Rate times Time). In our quadratic equation, the old D is now called X_final; the old R is V_initial. Our new distance equation is the next logical step. It not only looks at something's speed and the amount of time it travels, but also where it starts and whether the speed is changing (by the acceleration).

Going back to our bowling ball example, we determined before that the bowling ball was falling at a rate of 0 ft/s at the moment the ball was released (assuming you don't throw the ball downward), 32 ft/s after 1 second, and then 64 ft/s after two seconds, and so on. Our quadratic equation, however, finds distance; we can use it to find how far the ball has gone after a certain amount of time.


Calculation Time!
When we're using our equation to find the distance fallen (after some period of time), we need to know the values for all the variables besides the one that we're interested in. So when we drop the ball over the cliff, let's say that

• Initial position is Zero (since the ball has not traveled at all yet)
• which means that, for our equation, X_initial = 0
• Initial velocity is also Zero (since we're not throwing the ball either up or down)
• so V_initial = 0
• We know that acceleration due to gravity from the Earth is 32 ft/s²
• which write as A = 32

Plugging these values into our equation, X_final = X_initial + V_initial*t + (1/2)*A*t² becomes X_final = 0 + 0*t + (1/2)*32*t², which we can rewrite as:
X_final = 16*t²This means that we can determine how far the bowling ball has fallen by simply multiplying the time it spends falling, squared, by 16.

Doing some quick calculation:

• After zero seconds, the distance is Zero (the ball has not moved).
• After 1 second, the ball has fallen 16 feet
• X_final = 16*t² = 16*(1 sec)² = 16*1 = 16 feet
• After 2 seconds, the ball has fallen 64 feet
• X_final = 16*t² = 16*(2 sec)² = 16*4 = 64 feet
• And on and on and on... (of course, this all assumes that the ball doesn't hit the ground first)

We can even plot this if we want:


A parablola! [Note: Remember that the physical motion of the ball, though, is in a straight line down.]

Looking at the Math
The parabolic shape of the graph comes as no surprise to us, since we know that X_f = (1/2)*A*t² + V_i*t + X_i falls into the standard form of a quadratic equation
y = A*x² + B*x + CThis difference is that, in the equation for gravity, we use t as our independent variable and X as our dependent variable.

There are so many questions that can be asked at this point. What if we threw our bowling ball straight up first? What if we shot it from a cannon? And what if the cannon were pointed at an angle away from the cliff?



Instead of trying to answer all those questions right now, check out this applet about Projectile Motion (aka Shooting Stuff Out of a Cannon). Try it with different sized objects at several angles. After you've played with it for a while, try to find some patterns. And why does wind-resistance ruin your perfect parabola?!