



 
Hi Emily. How do you prove it is a parabola? Well, it depends on what you start with. If you can take as fact that the basic equations of projectile motion in physics are correct, then you can prove it algebraically. A projectile (such as a drop of water) ejected from the fountain has an initial velocity V_{0 } which has components V_{xo }and V_{y0 } which are horizontal and vertical components of the initial velocity. If you However V_{y } does change. It starts at V_{y0 } upwards (let's call upwards positive) and when the drop reaches the top of the arc, it has reduced to 0. Then it increases in magnitude, but becomes negative, until it The equation of projectile motion that describes the vertical distance (height) of the drop is just
where t is the time in seconds and g is the gravitational acceleration (negative) in distance per second. The equation of the projectile's horizontal motion is unaccelerated, so it is just x = V_{x }t. Notice that I am calling the starting point where the drop is ejected from the jet of water is the origin (y_{0 } = 0 and x_{0 } = 0). Now we have these two equations that describe what x and y are for a given initial velocity with respect to time. This is what we call a parameterized graph. So let's see if this is a parabola. If it is, we should be able to write it without reference to t, but with known quantities involved in the form y = ax^{2} + bx + c, where a, b and c are just scalar quantities based on known initial conditions (ie. the initial velocity components).
So since y = V_{y0 }t + (1/2)gt^{2}, we can substitute for t with the expression above.
which we can reorganize as:
Since this is indeed in the form of a parabola, this shows that the fountain arc is indeed parabolic. Emily, if you aren't allowed to use the basic physics equations of motion as starting points for your proof, you will have to try another approach. Experimentally, you could take a digital camera picture of a real fountain and then superimpose a graph of an appropriate parabola to show that it is "experimentally consistent" with a parabolic arc. But choose a calm day so that wind resistance doesn't affect your measurements! Stephen La Rocque.>  


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