3D Projectile Motion: Finding Y and Z Coordinates of Landing Position of Ball

[anmanc]

Homework Statement

A perfectly spherical ball is launched horizontally with a linear velocity of 9.9 m/s and an angular velocity (perpendicular to its trajectory) of 4.1 m/s. The ball's mass is 5.0g. (Diagram for explanation of coordinate system)
http://img221.imageshack.us/img221/2341/dddlcv.png

Find the landing position of the ball (its y and z coordinates).

b. If the rotational kinetic energy is increased by 10%, find the new landing position of the ball.

Homework Equations

Rotational KE = .5Iw^2
For a perfectly spherical projectile, I = 2/5 (MR^2)
Translational KE = .5mv^2
[PLAIN]http://www.sentynel.com/suvat.png

The Attempt at a Solution

For y coordinate, simply use v=x/t in x direction to find t - t=x/v=4.102/9.9=0.414s.
then, since in the y direction u=0 and a=9.8m/s^2, s=ut+.5at^2=0.83m above the ground on the wall.

I have no clue how to go about the z coordinate. Please help, someone.

 

[bp_psy]

Hint: There isn't any difference between motion in the x and z direction assuming Fz=0.

 

[anmanc]

But how can Fz equal zero if there is clearly a spin and angular velocity in its direction. Furthermore, this is a question to obtain a theoretical model based off of experimental data which clearly shows an angle phi (between the landing position and the launching device) in the z direction. This is what I am trying to figure out - how to calculate that deviation and perhaps the angle based on the data given. Is that possible?

 

[bp_psy]

Could you be more clear about what due you mean when you say "angular velocity"? You say that the angular velocity is 4.1m/s in the z direction. This does not make sense angular velocity is measured in rad/s.If the angular velocity of the ball is 4.1 rad/s in the z direction it means that the ball spins clockwise along the z axis. This is clearly not what you want.

What exactly is the launching apparatus?
Is there wind in the z direction?
What type of ball is it? Is it big enough to be influenced by the wind?

 

[anmanc]

Sorry for the lack of clarity.

I misused the terminology; I meant side spin speed. 4.1m/s is the speed of the side spin. Another calculation of the same concept that I have is just rate of spin, which is 17.8 rps again in the z direction. By the way, it is anti clockwise rather than clockwise.

The launching apparatus is a tennis ball machine. The machine has two horizontally placed wheels that have variable speeds; the ball is placed between them and launched through, so there is side spin (the ball's trajectory is rightward). The ball being launched isn't a tennis ball though, it is the equivalent of baseball without the stitching. There is no wind.

 

[bp_psy]

Sorry for the lack of clarity.

I misused the terminology; I meant side spin speed. 4.1m/s is the speed of the side spin. Another calculation of the same concept that I have is just rate of spin, which is 17.8 rps again in the z direction. By the way, it is anti clockwise rather than clockwise.

The launching apparatus is a tennis ball machine. The machine has two horizontally placed wheels that have variable speeds; the ball is placed between them and launched through, so there is side spin (the ball's trajectory is rightward). The ball being launched isn't a tennis ball though, it is the equivalent of baseball without the stitching. There is no wind.

I still don't completely understand your terms but I will try.
So initially the ball spins around your y-axis (a point on the ball would go towards positive z if you looked from the wall)?
The angular velocity of the ball is w=17.8rps=35.6pi rad/s. ( A measure of how fast it spins around it's axis of rotation)

Since the there isn't any wind the only force in the z direction would be the Magnus force but the problem doesn't appear to give enough info to find it.
http://en.wikipedia.org/wiki/Magnus_effect

 

[anmanc]

Thank you. Is there is no other way to calculate the z co-ordinate without this data?

But to see how it would be solved with the required data, let's assume the air resistance coefficient is 0.3.

 

[cjl]

The angular velocity will not affect the flight path unless you want to get into an aerodynamic model, which is fairly complicated. Without that factor, it's a simple projectile motion problem. With aerodynamics factored in, you can't just arbitrarily decide on an "air resistance coefficient". You could assume a coefficient of drag (which would probably be around 0.7), but the spin would still not affect the solution. For the spin to affect the solution, you would need to introduce a model with a coefficient of lift that is a function of the induced circulation of the fluid flow. That is a somewhat nontrivial calculation.