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

1. Aug 23, 2011

anmanc

1. The problem statement, all variables and given/known data
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 [Broken]

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.

2. Relevant 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 [Broken]

3. 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.

Last edited by a moderator: May 5, 2017
2. Aug 23, 2011

bp_psy

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

3. Aug 23, 2011

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?

4. Aug 23, 2011

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?

Last edited: Aug 23, 2011
5. Aug 23, 2011

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.

6. Aug 23, 2011

bp_psy

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

7. Aug 23, 2011

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.

Last edited: Aug 23, 2011
8. Aug 24, 2011

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.

Last edited: Aug 24, 2011