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# Parametric equations intro

Sal gives an example of a situation where parametric equations are very useful: driving off a cliff! Created by Sal Khan.

## Want to join the conversation?

• Other than a moving object in space, what are some real-life applications for parametric equations? • •   Physics formula. Yfinal=Yinital+VelocityInitial(T)+(1/2)A(T^2)
• • •   A parameter is some constant that relates two or more functions. In the example, the x-position and the y-position are not related to each other directly, but they are both defined in terms of time. Time is the parameter that allows us to see what the x and y functions are doing together. Try graphing the x and y functions separately. (In a calculator, you may have to call them y1 and y2, and change the t's to x's.) Can you look at them and know intuitively what the car is doing? Parametric equations allow us to break up a complicated problem, like motion in two or more dimensions, into simpler problems that can be solved separately and then recombined (if we want) through their shared parameter.
• at -- what makes Sal determine it's t2/2? Is this a formula? •  Well, I think the deduction of this equation comes out here:
d=Va*t, where d is the distance,and Va means the average velocity.
while Va=(Vf+Vi)/2, where Vf is the final velocity and Vi is the initial velocity (in this case Vi=0).
In addition,we know that the difference of velocity Vdelta=Vf-Vi=g*t. So,Vf=g*t+Vi,since Vi=0, so Vf=g*t+Vi=g*t+0=g*t.
Now replace Vf by g*t: d=Va*t=(Vf+Vi)/2*t=(g*t+Vi)/2*t=(g*t+0)/2*t=g*t/2*t=g*t^2/2.
• when will there be exercises for parametric equations? are you considering making them at all? It'd be really nice, thanks! • In "normal" functions (for the lack of a better word), like f(x)=x+2 for example, is "x" a parameter? • Sort of. It's tempting to say so, but parameter has a special meaning in this context. Each of the functions in the example are 'normal,' separate functions. What makes them parametric is that they share a parameter. Parametric equations are used when x and y are not directly related to each other, but are both related through a third term. In the example, the car's position in the x-direction is changing linearly with time, i.e. the graph of its function is a straight line. In the y-direction, however, its position is changing exponentially with time. The unifying 'parameter' is time. The car is moving through time equally "in both directions." This allows us to graph (x, y) coordinates to show the position of the car, as Sal showed. This is much more useful and intuitive than looking at the graphs of y(t) and x(t) separately. You can also use the parameter to find a unifying function that does directly relate x to y, as Sal hinted at.
Wikipedia has a pretty good blurb about the math uses of "parameter." http://en.wikipedia.org/wiki/Parameter#Mathematical_models
• There is a topic on the "Precalculus" mission called "Parametric equations and polar coordinates" but it has no skill excercises. Did they took them awary?? Or are there non?? Anyone knows?? Thank you very much. • I'm pretty tired so I may be looking too far into this... Is weight a factor? For example we figured out if this car is falling off a cliff at 5 m/sec sqrd it will land in the area sal figured out. Let's say that we had a tank or a marble traveling at the same speed with the same gravity acting on it, and our object is falling from the same height. I can't imagine the results will be the same. • The actual, real-world results are far more complicated because of the friction between the falling object and air as well as some minor issues with buoyancy. But, those kinds of computations are very advanced physics and engineering questions. They are just too complicated for this level of study.

I would anticipate, given its shape, that a marble would actually fall more quickly than a car. This would be because the car has an irregular shape and has lots of friction from the air slowing down its acceleration. A marble has a smooth, spherical shape, so it would have considerably less air friction. 