Ok so (UK) A2 Physics coursework time came, and I stupidly thought that projectile motion would be a fun investigation. But after just a little research I found it most challenging, it's taken me 'round the houses' via Reynolds numbers etc, and I've come to a halt as I'm kinda stuck.(adsbygoogle = window.adsbygoogle || []).push({});

I plan to see any relationship between the ideal, negligable-air-resistance trajectory of the projectile and its actual flight path. Perhaps compiling my own equation for the real range of the projectile, as an adaptation of the negilable air resistance equation:

[tex]x=\frac{2v^2\cos\theta\sin\theta}{g}[/tex]

therefore

[tex]x=\frac{v^2\sin2\theta}{g}[/tex]

My first question is more of a technical one that a calculation queery;would my capture method work?

I plan to set up a camera with a long shutter exposure in a very dark room. I would then set off a stobe light that would have a definate known flash frequency and I'd fire my projectile at the force and angle I wish. As the projectile (mainly spherical objects, perhaps other shaps when doing further investigation) takes it's flight, the light from the stobe reflects off of it and exposes it to the camera. Hopefully the light wont reflect much off of the dark background and will show the ball in it's flight at set times, enabling me to calculate initial velocity and the such.

So I'll be greatful for any comments on this.

Next, upon research I found that in most cases air resistance is assumed negligable which I feel would not give me much to write about and give my coursework an "AS" feel to it. So I am including it to give the investigation more depth.

I found that air resistance is sometimes proportional to velocity but it is more realistic to look at it as a square law, which also helps in gaining A2 marks.

As I learnt from AS, projectile motion can be split into two componants, verticle and horizontal.

First off, the horizontalvelocity would've been constant, but with an external force on it (drag) it will -negatively- accelerate and therefore won't be constant.

But as the projectile takes it's flight path diagonally up, across, then diagonally down, so the drag is always opposing is diagonally down then across, then diagonally up.(see fig 1. attached)

As the angle is constantly changes obviously the cosine of it constantly changes, therefore the horizontal componant of the drag (R in my diagram) changes, therefore it's non uniform acceleration...which as far as I'm aware may require some differential equations, I don't know how these are set up or used so I may need some help with that if possible.

As for theverticle componantof my projectiles trajectory, I split that up into two. Up and down,

For up, (as far as I'm aware)

initial velocity = [tex]v\sin\theta[/tex]

final velocity will be 0 at y(max)

acceleration either encounters the same problems as the verticle acceleration, or perhaps behaves as follows

[tex]\left(\frac{\frac{1}{2}\rho v^{2} A C_{d}}{m}\right)-g[/tex]

where rho is the density of air, v is the velocity (which of course is constantly changing due to this force so i don't know how that will work), A is the reference area, which apparently is related but not equal to the cross sectional area (not sure how to find that) and Cd is the drag coefficient.

Although I'm uncertain on how correct that formula is or whether it's applied in this situation.

As for thereturn flightdown to earth:

initial velocity would be 0

final velocity would be the objects terminal velocity which is given by the formula

[tex]\sqrt{\frac{2mg}{\rho A C_{d}}}[/tex]

correct? or would it not be it's terminal velocity?

and the acceleration would be the reverse of before

[tex]g-\left(\frac{\frac{1}{2}\rho v^{2} A C_{d}}{m}\right)[/tex]

Overall, the time taken will be the time taken for both parts of the verticle motion, and the range will be related to the time taken and the velocities/deceleration of the horizontal motion.

I'm unsure whether these equations are accurate or precise, and my main worry is how this differential equation thing is going to turn out. It's the only way I see around this constantly changing drag direction.

I'm not even entirely sure what im showing/finding/proving if I'm honest

So please tell me any comments, or anything ive done wrong or anything really

(and if you're wondering most of my background info came from wikipedia)

hope I ain't violated any rules, sorry about any spelling or latex errors, look forward to hearing from you lot, thanks in advance. ^_^

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# Projectile motion & drag

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