Maximizing Distance in Projectile Motion on an Inclined Plane

[confusedinMA]

Hey, I've taken Intro Physics years ago with an A but am reviewing some stuff and doing some of the harder problems out of Young and Freedman, I can't figure this one out to my embarrassment. Any help?

Heres the link of what I've done.

http://i.imgur.com/yD5G1.png"

 

[kjohnson]

Probably the easiest way to tackle that problem is to use a linear function (y=mx+b) that defines the ramp. Then look at your projectile motion equation for the vertical displacement 'y' as a function of time ( y=(1/2)a*t^2+v0*t ) where 'v0' is your initial velocity in the 'y' direction. The next two items are the main leaps in logic to solve this problem:

1. You need to make the first linear ramp function a function of time, not displacement 'x'. How would you make that adjustment?

2. You want to look for the moment in time when the ball hits the ramp. For this to be true what needs to be the relationship between the two functions (y values) at that time?

 

[kjohnson]

Sorry, I didn't really answer your question about maximizing the distance. To do so you need to take the derivative of the solution like you showed, buy using the product rule, chain rule...and set it to zero then solve. If you haven't taken calculus you can just graph the function and find the maximum value that way.