Ball thrown vertically upward deflection due to Coriolis force

[Ed Quanta]

Ok, so if a ball is thrown vertically upward with velocity v on the Earth's surface. (Air resistance being neglected). I have to show that the ball lands a distance (4wsin(beta)v^3/3g^2) to the west where w is the angular velocity of the Earth's rotation and beta is the colatitude angle.

fIhave equations for the centrifugal force where Fcf=-mw x (w x r)
for the Coriolis force where Fcor=-2mw x v

and through this should be able to solve this equation which goes something like

F=mg +Fcor +Fcf where F=mass multiplied by the acceleration of coordinate system. Help anybody?

 

[turin]

Are you supposed to solve or derive the equation that you have given in the first paragraph? It seems more like a derivation.

Off the top of my head, I would set it up in spherical polar coordinates. Don't forget that the equation for force that you gave is a vector equation. Basically, you will get 3 equations (that I think will be coupled). I see if I can give you more specifics if I find some time later.

 

[Ed Quanta]

Thanks, I have a lot of trouble with spherical coordinates and seeing 3 dimensions in my head and mathematically. I derived something sort of close looking using the vector equations I gave. Thanks for just taking a look at this anywayz.

 

[turin]

It seems like this is an exercise in choosing the reasonable level of approximation. For instance, you could go all out and solve in spherical polar coordinates, which would be a complete mess. You could take it down several notches and make some approximations like r ~ r₀ and θ ~ θ₀ throughout the process. I tried it this way, but I think I slipped up somewhere (I used Laplace transforms; it was still a mess, and I got a zero where I don't think there should be one). Something that just occurred to me, you could estimate the coriolis effect by applying it to the average value of the velocity (but that might just cancel/give zero). I might look at my approach again and see if I can find my error.