What is the formula that describes the vertical motion of a horseshoe pitched in the air from an initial height of 3 feet that reaches a height of 11 feet after 2 seconds and then falls to earth?

Answer 1

Before we begin, you should know that the 11-foot height achieved after two seconds is NOT necessarily the maximum height achieved by the tossed horseshoe. In fact, the question doesn't state that the 11 feet was reached on the way up. It may have been reached on the way up AND on the way down. In other words, was the shoe on the way up after two seconds or the way down? We don't know, but the equations may shed some light. The general equation for linear motion is d = d0 + v0t + (1/2)at2, where d0 and v0 are the initial distance (or displacement) and initial velocity, respectively, a is the acceleration, and t is time. For vertical motion, you can substitute the a in the above equation with the acceleration of gravity, g, which is -32.2 ft/s2. The value is negative because it is directed downward, which is often conventionally designated the negative direction. So, d = d0 + v0t + (1/2)at2 = d0 + v0t - 16.1t2. In this equation, the initial displacement is three feet, so d0 = 3. We don't know what the initial velocity, v0, is, but we can calculate it because we know that the horseshoe reaches a height of 11 feet at time, t = 2. Hence, your equation looks like this: 11 = 3 + v0 * 2 - 16.1 * 22. You'll have to use your run-of-the-mill algebra techniques to isolate v0. If you do it right, you'll get v0 = 36.2 ft/s. So, the general equation for this horseshoe toss is d(t) = 3 + 36.2t - 16.1t2. You can use that equation to figure out how long the horseshoe will stay in the air. Since d = 0 when the shoe hits the ground, just set the equation to zero and solve for t. It's a quadratic equation that will have two roots, but you'll be able to discard one of them as silly (because it will be negative). Since that equation is messy, you won't be able to factor it in your head, which means you'll need to use the quadratic formula to solve for t. If you do it right, you should get t = -0.0800248, t = 2.32847 seconds. Obviously, the first root is bogus, so t = 2.33 s.