Let's talk for a second about baseball. Space baseball (spaceball?). Imagine you're a pitcher situated on a mound deep in the vast expanse of space; what happens when you wind up and throw? With no gravity to pull it down, the ball you've thrown flies away at some constant velocity, sailing in a straight line forever - or at least until it comes close to another planet or star.
That's what the term bt does in this equation; it tells us that, in the absence of gravity, the baseball would sail upwards at a constant rate of b feet/second (or whatever units of length and time you happen to pick.)
Of course, we live on the Earth, and Earth pitchers and their baseballs are constantly pulled towards the ground by the force of Earth's gravity. This is what the -at² term does - it bends the would-be straight path of the baseball into an arc called a parabola, slowing its ascent and eventually bringing it back down to the ground (or hopefully into the umpire's glove!) The reason the sign is negative is because we're subtracting the pull of gravity from the ball's upward velocity every second.
So what does c do? Simple enough: it's the height we throw the ball from! If we're measuring in feet, throwing the ball from sea level would make c = 5 or so since the height of the average person is about 5 feet, while throwing it from the Empire State Building would crank c up to 1,250.
Now, what do our intercepts mean? The h-intercept is where our graph starts - when our time is equal to zero, in other words. As it turns out, the h-intercept is exactly the same as c! At the very beginning of the throw, the ball's height is exactly the same as the height we threw it from. The t-intercept, on the other hand, represents the point where the ball hits the ground; the further along the t-axis the intecept is, the longer it takes for that to happen!
Let's tweak c then, and see what happens.
If we throw the ball from a height of 5 feet, it'll climb to some maximum height, fall back down, and hit the ground after some total amount of time in the air.
Now, if we threw that same ball from the Empire State Building, we'd expect c to be a whole lot bigger. Here are some other things we could expect:
- Since we chuck it from much higher up, the maximum height reached by the ball will be much higher as well.
- The h-intercept - which, for the record, is the same thing as c (which, for the record, is the height we throw the ball from) - will be a whole lot larger.
- It'll take a whole lot longer for the ball to hit the ground since the ball has a lot further to fall, meaning our t-intercept will be much larger.
Looking at our options, we can see that changing c also changes the h-intercept, the maximum value of h, and the t-intercept.
