Date: 02/27/2001 at 22:20:45 From: D. Gilbert Subject: Setting up quadratic equations If a cannon is firing a projectile with an initial upward velocity of 100 feet per second, how do you set up a quadratic equation for the projectile motion of the cannonball it is firing? I know how to solve quadratic equations, but never thought about setting one up. Thank you for your help.
Date: 02/28/2001 at 10:57:08 From: Doctor Ian Subject: Re: Setting up quadratic equations Hi, Starting from first principles, we have Newton's law: a = F/m We can integrate acceleration to get velocity: v = (F/m)t + v_i And we can integrate velocity to get (the vertical) position: p = (1/2)(F/m)t^2 + (v_i)t + p_i In this equation, (F/m) is the familiar gravitational acceleration, g; v_i is the initial velocity; and p_i is the initial position. The problem tells you v_i: it's 100 feet per second. You can choose p_i to be whatever you want, although it's normal to set the location from which the motion begins as p = 0. And g is 32 ft/sec^2 in a _downward_ direction. If you're not familiar with integration, don't worry about it. I just did that to show you where the equations came from. In most problems of this type, 'setting up' just means choosing the right equation, e.g., p = (1/2)gt^2 + (v_i)t + p_i and going from there. Anyway, so now you end up with p = 100t - (1/2)(32)t^2 If you choose some height above the ground, you can solve for the time t that it takes to reach that height. (Note that there will be two solutions everywhere except at the top of the arc, since what goes up must come back down.) Or, if you choose some time of flight, you can determine the height of the cannonball at that time. Note that as time increases, the height eventually becomes negative, which makes sense if you think of launching the cannonball from the edge of a cliff: o o o o o p > 0 o o // ------- o p = 0 o p < 0 I hope this helps. Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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