I like Calculus, and it's comforting to know that practically everybody feels the same way about it. People love Calculus. And why not? It's challenging (the traditional authority figure euphemism for 'difficult') and requires not only a strong understanding of new concepts and procedures and the accuracy to complete complex problems without error, but solid algebra skills and a working knowledge of Geometry and Trigonometry as well. I mean, who doesn't have that?
One of the topics du jour is something called rectilinear motion, the one-dimensional change in position of an object. By one-dimensional I mean back and forth along a given line. This seems pretty removed from reality, but if you think about it, unless your car can drive sideways, fly or burrow, driving is pretty much one-dimensional business.
The core measurement of rectilinear motion is something called displacement, the change in position from where you started to where you end up. For example, driving from home to school, I have been displaced in some measurable kind of way. And even though I am three dimensional and so is my car, a road is kind of like a line, the one and only one-dimensional thing.
It is sometimes said that displacement is "distance with direction" and technically, that's true, but what it means is that moving in one direction cancels movement in the opposite direction. For example, if I'm lying on my couch and then get up and off to work I go and then home from work I go and then lie back down in the same spot on the couch, that my displacement is 0. But that's not a very good description of my day, which included my walking 100 ft to my car, driving 6 miles to work, then walking 50 ft. into the office, then back to my car, drive back home and get back on the couch. I actually traveled 12 miles in the car and 300 ft. walking. But my displacement is 0. Weird, huh?
So once you have displacement and, like it or not, displacement takes time, you have velocity, which is the change of position per unit of time. It has that same unreal directional component in that if my drive to work has a velocity of 30 miles per hour, my drive home is -30 miles per hour. And with velocity comes acceleration, which is the change in velocity per unit of time. This makes a little more sense in that if you're sitting still, then start moving, faster and faster until you reach the desired speed, then slow down until you stop again, that's positive and negative acceleration. Okay, but we're not up to the Calculus part of this yet, to be covered in the next post.
It is sometimes said that displacement is "distance with direction" and technically, that's true, but what it means is that moving in one direction cancels movement in the opposite direction. For example, if I'm lying on my couch and then get up and off to work I go and then home from work I go and then lie back down in the same spot on the couch, that my displacement is 0. But that's not a very good description of my day, which included my walking 100 ft to my car, driving 6 miles to work, then walking 50 ft. into the office, then back to my car, drive back home and get back on the couch. I actually traveled 12 miles in the car and 300 ft. walking. But my displacement is 0. Weird, huh?
So once you have displacement and, like it or not, displacement takes time, you have velocity, which is the change of position per unit of time. It has that same unreal directional component in that if my drive to work has a velocity of 30 miles per hour, my drive home is -30 miles per hour. And with velocity comes acceleration, which is the change in velocity per unit of time. This makes a little more sense in that if you're sitting still, then start moving, faster and faster until you reach the desired speed, then slow down until you stop again, that's positive and negative acceleration. Okay, but we're not up to the Calculus part of this yet, to be covered in the next post.
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