Is Air Resistance Negligible in Projectile Motion Experiments?

[Koin]

Homework Statement

I'm working on a simulated lab that measures the range and height covered relative to time for two objects in projectile motion fired at an angle on a leveled surface on the ground (so they're starting from a height of 0m), and there is air resistance acting on them (from the experimental findings its effect is minimal).
When I tried to find the theoretical acceleration for the objects I got the same acceleration for both objects even though air resistance is supposed to act on them differently (due to differences in mass, area, drag coefficient, textures, etc) and I also got the acceleration to be approximately -9.81, which is the value of gravity, and I thought this odd because air resistance is involved so it should have diverged from this value.
In the lab, the heavier object had the smaller approximation, which I got from making a graph on excel, finding the position equation, and differentiating twice, and this makes sense because acceleration is inversely proportional to mass. Also, the smaller object covered less horizontal distance and reached a lower maximum height and this also makes sense because light objects are more affected by air resistance.

Maybe why I am getting weird theoretical accelerations lies in the way I calculated it but I'm not sure conceptually what I did wrong. The relevant equations I used are listed. I would really appreciate any input on the process I used or my assumptions.

Thank you,
Koin

Relevant Equations

variables:
g= 9.81m/s^2
V = initial velocity or terminal velocity (I'm not sure)
Cd = coefficient of drag
r = rho or air density
A = cross-sectional area of object
Acc. for ascent: -g - (Cd*A*r*V^2)/2m
Acc. for descent: -g + (Cd*A*r*V^2)/2m
total acceleration for flight: the average of Acc. for ascent and Acc. for descent

Total acceleration equaled = -9.81

 

[BvU]

Hello Koin, !

Total acceleration equaled = -9.81

Please clarify what you mean with 'total acceleration' and how you come to his result of -9.81 ($\pm$?? ) m/s².

Did c_d come out substantially different from zero ?

 

[Koin]

Hello Koin, !
Please clarify what you mean with 'total acceleration' and how you come to his result of -9.81 ($\pm$?? ) m/s².

Did c_d come out substantially different from zero ?

Thank you BvU!

What I mean by 'total acceleration' is the acceleration for the total flight or path in the horizontal direction. I found this theoretically by first finding the accelerations for ascent and descent with the equations I listed and then finding the average of the two for each object individually. The Cd given for object one was 0.05 and for object two was 0.47. The cross-sectional areas for both objects were really small. The mass for object one was really small compared to the mass of object 2.

 

[Koin]

Homework Statement:: I'm working on a simulated lab that measures the range and height covered relative to time for two objects in projectile motion fired at an angle on a leveled surface on the ground (so they're starting from a height of 0m), and there is air resistance acting on them (from the experimental findings its effect is minimal).
When I tried to find the theoretical acceleration for the objects I got the same acceleration for both objects even though air resistance is supposed to act on them differently (due to differences in mass, area, drag coefficient, textures, etc) and I also got the acceleration to be approximately -9.81, which is the value of gravity, and I thought this odd because air resistance is involved so it should have diverged from this value.
In the lab, the heavier object had the smaller approximation, which I got from making a graph on excel, finding the position equation, and differentiating twice, and this makes sense because acceleration is inversely proportional to mass. Also, the smaller object covered less horizontal distance and reached a lower maximum height and this also makes sense because light objects are more affected by air resistance.

Maybe why I am getting weird theoretical accelerations lies in the way I calculated it but I'm not sure conceptually what I did wrong. The relevant equations I used are listed. I would really appreciate any input on the process I used or my assumptions.

Thank you,
Koin
Relevant Equations:: variables:
g= 9.81m/s^2
V = initial velocity or terminal velocity (I'm not sure)
Cd = coefficient of drag
r = rho or air density
A = cross-sectional area of object
Acc. for ascent: -g - (Cd*A*r*V^2)/2m
Acc. for descent: -g + (Cd*A*r*V^2)/2m
total acceleration for flight: the average of Acc. for ascent and Acc. for descent

Total acceleration equaled = -9.81

Also, sorry! I made a typo: instead of a smaller approximation, it should have said a smaller acceleration for the heavier object.

 

[haruspex]

V = initial velocity or terminal velocity (I'm not sure)

Since V occurs in your equations for the simulation you need to know what it represents. It's the speed at any instant, V=V(t).
You say the projectile is launched at an angle. You need to consider the drag in two dimensions. The magnitude of the drag depends on the speed, while the direction of the drag is according to the direction of the velocity. This means you need to calculate the magnitude of the drag and then resolve it into horizontal and vertical components to find the horizontal and vertical accelerations. You cannot consider the two coordinates completely independently.

But in regard to your observation that the simulated motion seems to suffer no drag, you need to look at the values you are computing for drag and how these compare numerically with the gravitational force. It sounds like they are much smaller, so you need to figure out why. Or maybe you just have a bug in your code.
If still stuck, please post your code and the numerical values for the parameters.

 

[Koin]

Since V occurs in your equations for the simulation you need to know what it represents. It's the speed at any instant, V=V(t).
You say the projectile is launched at an angle. You need to consider the drag in two dimensions. The magnitude of the drag depends on the speed, while the direction of the drag is according to the direction of the velocity. This means you need to calculate the magnitude of the drag and then resolve it into horizontal and vertical components to find the horizontal and vertical accelerations. You cannot consider the two coordinates completely independently.

But in regard to your observation that the simulated motion seems to suffer no drag, you need to look at the values you are computing for drag and how these compare numerically with the gravitational force. It sounds like they are much smaller, so you need to figure out why. Or maybe you just have a bug in your code.
If still stuck, please post your code and the numerical values for the parameters.

It's actually not my code. The simulation is by Univ. of Colorado's PhET. I also just realized that where I found the equations for ascent and descent acceleration (a NASA source) it said descent acceleration was 0 because after maximum height has been reached, the acceleration becomes 0 and terminal velocity takes over...so I guess I don't need to find the average and can just use the ascent's acceleration.

But it still differs significantly from the experimental acceleration. Excel gave me the y-position equations when I inputted the heights and times for each object individually. Then I differentiated twice to find the accelerations in the y direction. They are less than the theoretical accelerations I calculated for the ascents.

I'm thinking this could be attributed to the PhET simulation using a really low air density value as rho is not given in the simulation and for my calculations, I used 1.21...which means air resistance has a lesser impact in the simulation. Additionally, I tried to find the air density used by PhET by replacing 1.21 in my equation for ascent with x and setting the whole equation equal to the acceleration I derived from Excel. It was almost 88% less than 1.21. However, when I tried to find the air density for the second object the value was not consistent, the absolute value was less than 1.21 and it was negative.

Another thing I was considering was that differentiating the Excel-given position equation twice to find the acceleration is different from finding the acceleration of the ascent.

 

[haruspex]

descent acceleration was 0 because after maximum height has been reached, the acceleration becomes 0 and terminal velocity takes over

That makes no sense. It will descend some distance before approaching terminal velocity. Please post the link.

Excel gave me the y-position equations when I inputted the heights and times for each object individually.

What heights and times? Did you get those from the PhET simulation? What were your inputs to that? Maybe post a link for that too.

less than the theoretical accelerations I calculated for the ascents

How did you calculate those?

 

[Koin]

That makes no sense. It will descend some distance before approaching terminal velocity. Please post the link.

What heights and times? Did you get those from the PhET simulation? What were your inputs to that? Maybe post a link for that too.

How did you calculate those?

https://www.grc.nasa.gov/www/k-12/airplane/flteqs.html
That's where I got that information on acceleration for descents from.

And yes I got the heights and times from the PhET simulation:
https://phet.colorado.edu/sims/html/projectile-motion/latest/projectile-motion_en.html
to see the lab playout click the box for lab,
select the object (my first object was a football)
keep the diameters and masses how you find them
check the air resistance box
set the angle mine was (45 degrees)
set the speed mine was (15m/s)

To find the ascents I used one of the relevant equations I listed in my original post:
Acc. for ascent: -g - (Cd*A*r*V^2)/2m

 

[Koin]

Since V occurs in your equations for the simulation you need to know what it represents. It's the speed at any instant, V=V(t).

Also if you mean instantaneous velocity I’m not sure how effective incorporating drag force into my calculations for acceleration is going to be because doesn’t that mean I would have to find the velocity at each second (time was given in seconds) and repeatedly update the velocity to update the acceleration. I guess this just goes to show that acceleration isn’t constant throughout the projectile motion with air resistance involved and if so what exactly is the value I got from making a trendline on Excel for the heights and times and finding a position equation and then differentiating twice?

 

[haruspex]

That's where I got that information on acceleration for descents from.

... where it does specify light objects. The lighter the object, in relation to its cross section, the sooner it will near terminal velocity, but it is still not immediately on starting descent.

select the object (my first object was a football)

I note that that selection sets the drag coefficient to 0.05. That is much too low. See e.g. https://en.m.wikipedia.org/wiki/Drag_coefficient, where 0.5 is nearer the mark for a sphere. For pumpkin, PhET more reasonably assumes 0.6.
So I tried selecting pumpkin but putting the mass as 1kg, the lowest it would allow, and the diameter as 0.27, so that the mass to area ratio was about the same as for a football. The difference between with and without air resistance was a lot starker.

 

[BvU]

Also if you mean instantaneous velocity I’m not sure how effective incorporating drag force into my calculations for acceleration is going to be because doesn’t that mean I would have to find the velocity at each second (time was given in seconds) and repeatedly update the velocity to update the acceleration. I guess this just goes to show that acceleration isn’t constant throughout the projectile motion with air resistance involved and if so what exactly is the value I got from making a trendline on Excel for the heights and times and finding a position equation and then differentiating twice?

Why don't you show us your work, so we can can compare with what we 'observe' ?

if you are interested, the equations here and the spreadsheeet
http://galileo.phys.virginia.edu/classes/581/Projectile5.xls give you more control

 

[Koin]

... where it does specify light objects. The lighter the object, in relation to its cross section, the sooner it will near terminal velocity, but it is still not immediately on starting descent.

I note that that selection sets the drag coefficient to 0.05. That is much too low. See e.g. https://en.m.wikipedia.org/wiki/Drag_coefficient, where 0.5 is nearer the mark for a sphere. For pumpkin, PhET more reasonably assumes 0.6.
So I tried selecting pumpkin but putting the mass as 1kg, the lowest it would allow, and the diameter as 0.27, so that the mass to area ratio was about the same as for a football. The difference between with and without air resistance was a lot starker.

I talked with one of my TAs and he explained that basically the lab was configured to show that air resistance was still pretty much negligible. This makes more sense than trying to compute the acceleration with drag force factored in as the acceleration is not constant throughout the whole path.

Thank you so much for your for your help!

 

[Koin]

Why don't you show us your work, so we can can compare with what we 'observe' ?

if you are interested, the equations here and the spreadsheeet
http://galileo.phys.virginia.edu/classes/581/Projectile5.xls give you more control

I actually saw that spreadsheet but I wasn't sure if it would be a close simulation to PhET and I was pressed for time to submit the lab report. Also, the reason I don't want to share my work is that I have to submit it and it might go through something like Turnitin to check if students are plagiarizing. I just don't want to risk it.
Thank you for your insight! Truly every response I got helped sort of lead me to something conclusive and a final resolve.
Thanks again!

 

[haruspex]

I talked with one of my TAs and he explained that basically the lab was configured to show that air resistance was still pretty much negligible. This makes more sense than trying to compute the acceleration with drag force factored in as the acceleration is not constant throughout the whole path.

Thank you so much for your for your help!

I still say the 0.05 drag coefficient for a football is too low by a factor of 10. The coefficient is determined by the overall shape, a sphere (or ellipsoid?) in this case, and surface texture, such as dimpling.