Solved: Solving Ball Kicked from Top of Hemisphere Problem

[phillipsc]

Homework Statement

A person standing at the top of a hemispherical rock of radius R kicks a ball (initially at rest on the top of the rock) to give it horizontal velocity Vi. What must be its minimum initial speed if the ball is never to hit the rock after it is kicked? With this initial speed how far doesit go from the base of the rock when it hits the ground.

The Attempt at a Solution

The problem I'm having with this is that they don't initialize any variables. tey don't tell you R or the time it took. They only say that it is horizontal. the problem is a 2d planer by the way. So i realized that at least the magnitude of the ball must always be greater than the magnitude of the radius at any x position, otherwise the ball hit the circular rock.

I'm probably just short of a logical comprehension of the question. does anyone know how i might be able to solve this. It seems like the problem isn't really plugging values into formulas but more logical.

i visualized the problem as the balls initial position is ( 0,R ) when R is the radius and the origin is the middle of the hemispherical rock. Then the final position of the rock would be (D + R, 0 ) D being the distance from the base of the rock to where it landed."

The only other piece of the puzzle would be that the ball falls at a -9.8 m/s^2 acc. because of gravity obviously. any input or help is GREATLY appreciated.

 

[sahana.r]

hey
I'm having problems to this same question. Did you ever find the solution?

 

[Avodyne]

Welcome to the forum!

Forget the rock for a moment. At time t=0, the (x,y) coordinates of the ball are (0,R). The initial horizontal velocity is vi. The initial vertical velocity is zero. Can you find y as a function of x? (If not, try first finding x as a function of t and y as a function of t, and then eliminate t.)

Now, you have y(x), the height of the ball as a function of horizontal distance. What is the height of the surface of the rock as a function of horizontal distance? Let's call it z(x). You want y(x) > z(x) for 0<x<R. How big must vi be for this to be true?

By the way, by dimensional analysis, the answer must be of the form vi = (number) g/R. This is because g (the acceleration of gravity) and R (the radius of the rock) are the only constants in the problem, and only the combination g/R has dimensions of velocity.

 

[cherry_ying]

I'm having the same problem with that question. I have tried the method you suggested, however, it just results in a big messy equation (to degree 4?) that doesn't solve.

 

[Avodyne]

What did you get for x(t) and y(t)?