How Does Air Resistance Impact Projectile Motion in Sports?

[hidajua]

Homework Statement

I have started my extended essay for the IB, My REsearch Question is "How does air resistance affect the projectile motion of sports balls?" i am collecting data by video taping the throws and measuring where the ball is in every frame. My supervisor tells me that i have to have and equation so that i can see the difference between the actual throw and the throw without air resistance, i don't now how to get that equation.

Homework Equations

v² = u² + 2as, S = ut + 1/2at², v = u + at, d = vt

The Attempt at a Solution

I tried to generate a best fit equation of all the values collected, and then a best fit equation for the first values (since those values are the least affected by air resistance) and compare both equation's positive x-intercept

 

[jambaugh]

I'm not sure how much detail you are being asked to give. Are you only looking at the horizontal component of motion?

If so then neglecting air effects the horizontal motion should be linear. This is your last equation:
d = v \cdot t
where d is the horizontal position (taking the initial position as 0) and v is the initial velocity.

(Note v in this last equation should equate to u in the others.)

As you mention you should estimate the initial velocity from the first few frames.
Speed = (distance / time) = ( distance times frames-per-second / number of frames).
This will be the same initial velocity ( u ) used in the equations which are taking air resistance into account.

The other equations all appear to be assuming acceleration is a constant. Wind resistance is not a constant but depends on speed. You can use these equations however to get a first order approximation if the ball is not slowing down too much throughout your record of each throw. If the initial speed of various recorded throws varies by much you ought to plot acceleration as a function of this varying speed.

The first equation:
v^2 = u^2 + 2a\cdot s
comes from the energy and work done on the ball by the air assuming the acceleration (and force) are constant. It appears as if v is the varying velocity and u the initial velocity in this equation. Note that if you multiply by m/2 and apply force = mass x accel you get KE = Initial KE + Work done. (KE= Kinetic energy = m/2 velocity squared)

The second and third equation we get from calculus assuming constant acceleration. S appears to be the horizontal position at time t and again u is the initial velocity and v appears to be the variable velocity (as a function of time). Again the acceleration a should be taken as a negative values since it opposes the direction of motion (which I assume you want to take as positive).

If you really want to get into more detail you should put the position for each frame into a column in a spreadsheet. Then in the next column take the differences times the frames per second to get the velocity. Then in the third column again take the differences in velocities times the frames per second to get the acceleration.

Do this with several trials and then try to find the best fit for acceleration as a function of the velocity. Try researching online to find the best type of function to use. (I could tell you ... but then I'd have to kill you )

If the accelerations don't change much over a single video then you could alternatively just get the average acceleration and do trials for different speeds to see how acceleration depends on speed.

At the very least you should plot position and velocity as a function of time comparing the constant velocity (no acceleration) case with the actual data (with equivalent initial velocity).

You should also at least mention that wind resistance depends on the speed. A good way to end your essay is to explain how this affects your results and suggest improvements based on this fact.

I hope this has been helpful.