Projectile motion of tennis ball with air resistance

[Shruf]

Hi, so I am trying to model the path of a tennis ball when serving. I already have the model without air resistance, but now I'm getting into differential equations with the air resistance. I obtained two differential equations for acceleration that i think are correct, but I'm not sure where to go from here exactly. It is in two dimensions. What I want to find is what angle the ball must be hit at to travel a set distance.


So far these are what i have:
equation 1: m[itex]\stackrel{d^{2}x}{dt^{2}}[/itex] = -k [itex]\stackrel{dx}{dt}[/itex] [itex]\sqrt{\stackrel{dx}{dt}^{2}+\stackrel{dy}{dt}^{2}}[/itex]

equation 2: m[itex]\stackrel{d^{2}y}{dt^{2}}[/itex] = k [itex]\stackrel{dy}{dt}[/itex] [itex]\sqrt{\stackrel{dx}{dt}^{2}+\stackrel{dy}{dt}^{2}}[/itex] - g

K is drag co-efficient, m is mass, g is gravity acceleration, all of which are defined.I have the initial velocity, and I know that I can use trig ratios to get the initial x and y velocities, but I have not idea what to do with them. BTW all the above below things are meant to be fractions, but i am not sure how to make them fractions, sorry.

 

[haruspex]

Hi, so I am trying to model the path of a tennis ball when serving. I already have the model without air resistance, but now I'm getting into differential equations with the air resistance. I obtained two differential equations for acceleration that i think are correct, but I'm not sure where to go from here exactly. It is in two dimensions. What I want to find is what angle the ball must be hit at to travel a set distance.


So far these are what i have:
equation 1: m[itex]\stackrel{d^{2}x}{dt^{2}}[/itex] = -k [itex]\stackrel{dx}{dt}[/itex] [itex]\sqrt{\stackrel{dx}{dt}^{2}+\stackrel{dy}{dt}^{2}}[/itex]

equation 2: m[itex]\stackrel{d^{2}y}{dt^{2}}[/itex] = k [itex]\stackrel{dy}{dt}[/itex] [itex]\sqrt{\stackrel{dx}{dt}^{2}+\stackrel{dy}{dt}^{2}}[/itex] - g

K is drag co-efficient, m is mass, g is gravity acceleration, all of which are defined.I have the initial velocity, and I know that I can use trig ratios to get the initial x and y velocities, but I have not idea what to do with them. BTW all the above below things are meant to be fractions, but i am not sure how to make them fractions, sorry.

Assuming drag is proportional to the square of the speed (which is a fair approximation at high speeds) then your equations are basically right. You do have the sign wrong on the drag term in the y-equation. Drag will always oppose the direction of movement.
For the LaTex, put the m inside the LaTex, use \frac, not \stackrel, and wrap the dx/dt terms in braces so that the squaring applies to the whole term, not just the x or y:
[itex]m\frac{d^{2}x}{dt^{2}}[/itex] = -k [itex]\frac{dx}{dt}[/itex] [itex]\sqrt{{\frac{dx}{dt}}^{2}+{\frac{dy}{dt}}^{2}}[/itex]
As to solutions, I believe there is no closed form solution.

 

[ehild]

you can solve the problem with linear drag. http://en.wikipedia.org/wiki/Trajectory_of_a_projectile

ehild

 

[Shruf]

As to solutions, I believe there is no closed form solution.

Thank; I do know that there is no closed solution but I am wondering how to do the mass calculations on something like excel? I am not sure how to incorporate these two equations for acceleration in my position equation, or in my calculations for range. Also, I think my sign is correct since the ball is being hit downwards, which I've made negative. I think i have to integrate the equations, but I'm not sure in what way I need to use them yet.

 

[ehild]

See

https://www.google.com/url?sa=t&rct...dBNfruPKIOjic-g&bvm=bv.58187178,d.Yms&cad=rjt.

You find the the two-dimensional problem at the end, with the suggested method of solution.
Read also

http://www.google.co.uk/url?sa=t&rc...-NA6KuuNWFpj2JQCA&sig2=3ba5u2OYcoq60dP87Bejqw


ehild

 

[haruspex]

I think my sign is correct since the ball is being hit downwards, which I've made negative.

The contribution of drag to the acceleration must be a negative multiple of the speed.