Projectile Problem

[sidius]

From the 2011 Study Guide, 09H-37:

Erica stands 20 ft away from a 10 ft tall wall. She throws a ball at a release height of 4 ft that just clears the wall. What is the ball release velocity?

The answer is 38.2 fps

Working backwards using Dh=[(v^2*sin2(theta))/gravity], it looks like release angle theta=30.96 deg and Dh=40 to achieve the correct answer of v=38.2 fps. The release angle makes sense to me, but why would horizontal distance be 40?

[SumitG]

The range equation accounts for the entire parabolic projectile motion, whereas in this problem, only half of the parabolic motion is completed. This is why the range is actually [unparseable or potentially dangerous latex formula].

[sidius]

That makes sense, thanks!

Do you have a step-by-step solution to the problem? I think I've got it,
but I don't really trust it for when it pops up on the test w/ different numbers.

[SumitG]

Sure; actually, you've got most of the solution already. The only thing extra you need is a vertical displacement equation. Then, you just need to solve the horizontal range equation for [unparseable or potentially dangerous latex formula].
Starting with the horizontal and vertical range equations, we have

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

Since you already know [unparseable or potentially dangerous latex formula], you have enough information to solve for [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]. Dividing both equations, you have

[unparseable or potentially dangerous latex formula]

and hence

[unparseable or potentially dangerous latex formula]

Then, you can just plug back into one of your original horizontal or vertical range equations and solve for [unparseable or potentially dangerous latex formula].

[unparseable or potentially dangerous latex formula]

[mathmajor]

Is there a place where you can find the study guide on line?

[sidius]

Yes. Dr. Bourell puts the Study Guides and Solutions on the official Calculator Contest page annually. Look under "Study List PDF Files":

[mathmajor]

Thanks!