# Determining speed experimentally

#### TOD

1. Homework Statement
Question:
What are some ways in which one can determine the velocities of a projectile experimentally other than the multi-flash camera in a dark room?

Background info:
I'm working on an essay about air resistance upon a shuttlecock. Now, before I start doing any maths, I need some raw data on the motion of a shuttlecock when hit by a badminton racket. The problem is, I'm a beginner and I don't have or know the equipment suitable for determining the velocities of the shuttle cock. Now I've thought about doing this in a darkened room with a multi-flash camera, but that'd be kind of hard since the range for the shuttlecock is longer than any dark room I'm available to. Especially if I also want to show where the net and the court is so I could also analyse the experiment/data with respects to badminton as a whole. So I'd need to move the multi-flash camera to a badminton court at night. And that a lot of other problems arise.

2. Homework Equations
v=d/t!?
F(d) = (1/2) C(d).A.rho.v^2

3. The Attempt at a Solution
Yeah, I'm looking for one. :grumpy:

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#### denverdoc

well the simplest soln is to subject it to constant acceleration, that is drop the shuttlecock from various heights and measure time.

#### TOD

But acceleration isn't constant, especially with a shuttlecock and especiallys since I'm looking at an inconstant force acting opposite to the gravitational force of attraction. So I'm more looking to something that can record distance, velocity, time, whatever that can help me find the instantaneous (or very close to) velocity at a known time after release/hit by a racket.

#### denverdoc

The force f gravity=ma, is constant. What varies is the force due to drag which in this case is proportional to velocity squared. Timing a descent from various altitudes allows a determination of the frictional resistance. Given that you can predict much about its behavior under many circumstances.

#### mezarashi

Homework Helper
The kinematics equations don't apply when the acceleration is non constant as with a shuttle cock. You need to fall back to the fundamental equations the kinematics equations were derived from, that is:

$$a(t) = \frac{F(t)}{m}$$
$$\frac{dv}{dt} = a(t)$$
$$\frac{ds}{dt} = v(t)$$

in this case F is a function of v(t). t represents the variable being a function of time. If you can write equations for the force on the cock at anytime, then you can write an equation for its acceleration at any time. Integrating up, you can get velocity and displacement.

#### denverdoc

We seem to be getting tangled up a bit, I never suggested that the actual acceleration was constant, only that gravity was, hence:

ma=mg-K*v^2

#### TOD

Which is also what I'm trying to do. If I can find the acceleration of the cock at any time by experiment, I can find the force of drag (since i can find out the weight force of the cock) at those certain times.
Now the other problem with integration to find the range of the shuttlecock is that the force is inconstant at an inconsistent way! The drag force depends on velocity squared and velocity is dependent on the net force which is partly dependent on the drag force! I'm screwed.

Look, I'm just trying to start my essay. The rest I'll just do as I go. The problem now is finding an apparatus that can help me find velocity at certain points in time.

#### denverdoc

Not inconsistent, just not your average bear, you can convert above into

v'/(k/m-v(t)^2)=0, and use partial fractions to solve, or trig subs.

#### TOD

utterly confused by taht equation, but if it helps to determine the range of the shuttlecock (which is the final aim of the whole essay) at given the initial velocity and angle then I better understand it.
Still, how do any of those equations help me come up with an experiment that will record the displacement/velocity?

How did the old physicists find velocity (which they often need in order to do any calculations) without a multi-flash camera!?

#### mezarashi

Homework Helper
Absolutely precise calculations involving drag are complex if not impossible to solve theoretically. Generally, computer simulation is used because of the complex effect you get at high velocities (ie. when you need to include the drag coefficient).

If however, you plan your experiment well enough, you can use the much easier equation for viscous drag (http://en.wikipedia.org/wiki/Drag_(physics)#Drag_at_low_velocity.3B_Stokes.27s_Drag) with force being simply a function of velocity. This type of problem is a regular exercise in Introductory Differential Calculus classes (topic: ordinary differential equations) where one must solve for the velocity function.

Wouldn't experimentally determining velocity mean directly measuring it? Whether it be by camera or through some ultra-sonic detector. Sorry, actually I'm a bit confused about what you want to achieve at this point. I feel that I've sidetracked a lot.

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#### denverdoc

utterly confused by taht equation, but if it helps to determine the range of the shuttlecock (which is the final aim of the whole essay) at given the initial velocity and angle then I better understand it.
Still, how do any of those equations help me come up with an experiment that will record the displacement/velocity?

How did the old physicists find velocity (which they often need in order to do any calculations) without a multi-flash camera!?
Just subbing dv/dt for a in Newtons second. ie a=dv/dt and equating with the two forces of gravity and air resistance acting in the SC.

But your real question if I understand, has to do with how the old guys did it. Well the answer is still in the eqn that relates velocity to distance & time. If you have an object whose speed varies with velocity as a shuttlecock, there is no easy method except to solve the eqn I posted and plug in values to determine k.

Now here it gets even messier, at some speeds and viscosity of the media the SC is traveling thru, it may depend more on velocity than velocity squared. Intuitively this is a matter of inertia vs drag force. A feather has little inertia, lots of drag, and a cannonball the opposite.

In the end it comes up with measuring displacement and time, unless one could say hook a whistle to the SC and use doppler shift or equate energy in meaningful way,

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#### TOD

Ok, this is what I have in mind.

I am writing my extended essay (IB) on badminton.

Firstly I'd like to find the coefficient of drag by measuring the terminal velocity (dropping cock vertically down) with a multi-flash camera and then subbing into the drag equation. I really want to use the drag equation at high velocity since badminton is the 'fastest' racket sport. I'm not sure though, maybe the 'viscous drag' can help me.

Then after knowing the coefficient of drag, I'm hoping that it's the same in when working in the horizontal dimension when the shuttlecock becomes a projectile (under two forces in both directions). Now I want to use all the information I have to determine the time for the cock to reach the ground in the vertical sense, then sub in the time into the horizontal equation (like u would when working out the range with only 1 force) and work out the range with air resistance.

I don't mind doing a lot of maths and physics since that's what I want to do and I have 4000 words to fill up anyway.

#### denverdoc

Good thinking. methinks you needed no help after all. Well done!

#### TOD

However, my question remains un-answered. I need an apparatus other than the multi-flash camera, which will be troublesome to borrow from school and move around, to determine the varying velocity in the horizontal dimention with a certain initial velocity (also need to be found by the experiment). So that I can sub in the initial velocity and the angle of launch into the equation I (hope) will come up with near the end of the essay and find the range.

If this makes things any clearer, here is my essay topic (subject to change):
Determining a Model for the Range of a Shuttlecock at Known Initial Velocity and Angle

Remebering that the shuttlecock travels in an almost-parabolic curve in serving in a badminton game.

SO once, again, my question. What kind of apparatus can help me find the velocity/speed!? I mean, before the multi-flash camera was invented, people must have had ways to determine the instantaneous speed to prove theories or come up with theories and all that. What is it that they use? Or how did they do it without such an equipment? Say in projectile motion, how did they prove that the theory behind projectile motion (motion can be separated into two, or sometimes more, dimentions) works?

#### denverdoc

a simple video camera will do.

#### TOD

I've thought about a video camera, but the quality is not as good as a picture camera... Plus, I can't put a video on my paper. and explaining what I've done would take up 2000 words... Pointless talk, so yeah. I think Multi-flash would still be better than a video. Which still doesn't solve my problem of choosing something other than the multi-flash camera. How did the old guys do it!? They obviously didn't have a video camera back then either!

#### denverdoc

What about a video with a grid in the background? Here you need to convert the video to still frames and since they occur every so often a perfect map of motion vs time. With a bullet good luck!

#### TOD

Well, thanks for all your unconditional help everyone. I think I've got some new ideas now even though I still don't know what to do with the apparatus. Yeah, I guess we'll stop here now.

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