posted
I'd like to create an activity for Algebra II students where they measure vertical and horizontal velocities of projectiles (like, say, a ball )and fit equations to arrive at the formulas for this sort of motion, x = v (x) t and y = v (y) t – 16t^2, and also to be able to make predictions about how far a projectile will travel along each vector and how long it will remain in the air. They can't derive the formulas using calculus.
I could give them the formulas, and just have them record measurements, but it seems more interesting to have them discover the forumulas themselves.
I think I might be able to borrow equipment that will measure speed, but it is possible to measure velocity only in one direction? In other words, a ball may be traveling at a 45 degree angle . . . can I measure it's horizontal velocity and its vertical velocity as components of the general velocity?
(I realize this isn't too well thought out yet. This is still the early stages of an idea.)
posted
Something that would be easier would be to use the y formula to figure the vertical component of velocity based on how long it takes an object to hit the ground after being thrown. But then I have to give them the formulas first.
Posts: 13680 | Registered: Mar 2002
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posted
x being the angle of elevation of the thrown ball?
That's a good idea. A couple of challenges, though: measuring the angle, and these kids don't know trigonometry yet. That doesn't rule it out, though. I could teach them just enough to use it. But my goal is for as little as possible to come from me, and for as much as possible to be discovered through experience.
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posted
You could video tape the flight and advance the tape a few frames at a time while the students make measurements on the screen. You could probably even determine the scale between the image on the video and the actual equipment so everything is correct.
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posted
That is exactly what I was thinking. Shoot the ball with a grid behind it. Go forward frame by frame and read off the grid how far the ball has moved from frame to frame.
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posted
Yes, I remember the video thing from one of my physics classes. And yes, the length of time between frames is standard (equal).
Posts: 1813 | Registered: Apr 2001
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Shoot the ball with a grid behind it . . . or superimpose a grid on the image afterward? And would I be getting accurate measurements of initial velocity?
Thanks for all the ideas so far, btw. Posts: 13680 | Registered: Mar 2002
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posted
Include a running timer (borrow one from your local olympic-sized swimming pool), to give you seconds and fractions of seconds at each frame.
Video typically runs at 30 fps in the US, and 25 fps (I think) where you've got 50 hz power. Film runs at 24 fps (sound) and 18 fps (silent).
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posted
For a more low-tech solution, shoot your ball along a blackboard, and have kids standing ready to mark its height with chalk as it goes by them. The points should trace a parabola.
If you want serious high-tech, you might be able to get laser equipment to measure speed from the Doppler effect. I know such things exist even for school labs, though I'm not sure about the price. That will measure the speed along the axis you're pointing your laser.
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quote:If you want serious high-tech, you might be able to get laser equipment to measure speed from the Doppler effect. I know such things exist even for school labs, though I'm not sure about the price. That will measure the speed along the axis you're pointing your laser.
I'm pretty sure I can get access to stuff like this. It measuring speed along the access I point it makes sense, but then, if I want vertical speed, I would have to point it vertically, right? If the object is moving along a horizontal access as well as a vertical one, wouldn't I then have to keep the measuring equipment under the projectile somehow?
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posted
You could fake it. Have the appropriate numbers or formulas to plug the numbers into and then give the kids the result. Saves all that messy needing to measure crap.
Posts: 10177 | Registered: Apr 2001
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posted
The "messy needing to measure crap" is kinda what I'm going for. Every once in a while it's good to get outdoors, and let 'em get their hands dirty with math, so to speak. I don't know how strong the actual educational benefit is, but if nothing else, it serves a nice motivational benefit. The kids perceive math as a class where they do occasional interesting activities, and that helps get them through the less hands-on times.
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posted
Oh no, Ick, I'm not saying don't do the activity. It's sounds cool. What I'm saying is, fake up a machine looking thing that you say is going to do the measuring and then give them what would be the right measurements.
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quote:I'm pretty sure I can get access to stuff like this. It measuring speed along the access I point it makes sense, but then, if I want vertical speed, I would have to point it vertically, right? If the object is moving along a horizontal access as well as a vertical one, wouldn't I then have to keep the measuring equipment under the projectile somehow?
You're mixing your 'axis' with your 'access' there. But yes, having to keep the equipment under the object is one problem with the method. You might try having several of your laser thingies, point them straight up, and measure the speed at different points.
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posted
No, good gracious, heavens no. I would never advise someone to bamboozle their students. No, I'm merely suggesting that Icarus hoodwink them, which, as we can all agree, is an entirely different kettle of fish.
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posted
I know at college there were two different versions of th physics classes. One with calculcus and one without. Maybe you could look into getting one of the books without to get the proper equations and lessons.
Posts: 2845 | Registered: Oct 2003
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posted
I know the proper equations. What I'm looking for is to get kids to discover them. You do bring up an interesting point, though. I wonder what sort of lab activity a college elementary physics class would have on this. Unfortunately, my physics class did have calculus as a requirement, and I don't have any old lab materials anyway. Do any of you still have an old lab book or something?
posted
An alternative to the video solution is to use a still camera and a strobe. You can easily calibrate the strobe timing, and since all the images of the projectile appear on the same frame, you can see the parabolic path in one image, instead of piecing it together from multiple frames.
Does anyone besides me still own a fully manual SLR?
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posted
My physics class here at Drexel used the strobe method. (It was w/o calculus version, as you have to be an engineer or take beyond the first year of physics to get a class w/ calculus, for whatever reason.)
Posts: 1621 | Registered: Oct 2001
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Strobe photoraphy would also work -- grid behind, strobe flashing at known frequency...
Another thought that requires NO special measurement equipment, and might be simple to implement:
Build a "ball launcher" -- essentially an inclined ramp that will shoot the ball out at the same velocity and initial angle every time.
Use a tennis ball.
Wet the ball so when it hits the wall, it makes a mark. (wet the ball thoroughly each time for a standard amount of time).
Take measurements of the ball's impact height on the wall when you shoot it from an array of distances.
If you plot height over distance, you'll get a parabola -- approximately, but probably close enough if you do the measurements 10x at each selected distance. Something like that.
Now, to time the speed of the ball, have kids measure it as follows:
Two stop watches.
One kid yells "go" as the ball is released at the top of the launcher. Both kids start their stop watches.
Another kid yells "stop" as the ball leaves the launcher. Stop watch 1 is stopped at this point.
A third kid yells "stop" as the ball hits the wall. (or the kid listens for the ball to hit the wall and hits "stop"
Over successive trials at a fixed distance, an average difference in the two times will be obtained. Knowing the distance between the launcher and the wall, you have distance traveled. the difference in the times for the two stop watches is the time it took (on average) to travel that distance.
distance/time = speed.
You could try this at various distances to demonstrate that the speed in the horizontal direction is pretty much constant as long as the ball is in the air -- within measurement limits.
If friction of the "wet" ball is a problem, you could try this with a rubber ball that has been lightly inked and then hang a piece of paper on the wall where the ball is going to hit.
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posted
Got a problem with stopwatches. Say ya got an 8foot(2.44metre)ceiling. Drop a ball from the ceiling, and it'll hit the floor in ~0.7seconds. Human reflexes are in ~0.2second range. Take one person yelling "stop" and another hitting the stopwatch, and ya got a ~0.4second margin of error. Which is a heck of a large portion of 0.7seconds.
Anything much more accurate is due to the natural mathematical processing by the brain to anticipate where&when a moving object is going to be, and acting on that anticipation before the drop and before the hit.
What goes up must come down also applies to the time: ie the time a ball takes to drop from the ceiling is equal to the time it takes to reach the ceiling. If the ball takes less time to reach the ceiling, it bounces off the ceiling; which messes up the point of the experiment. So any launch-to-landing in an 8foot tall room will take a maximum of ~1.4seconds, irrespective of horizontal velocity; provided that the ball doesn't bounce of the wall (compression, friction, etc).
And the above is a "let us start with a spherical cow" example. Your classroom isn't going to be a vacuum, the ball isn't going to be inertialess, and the ramp&ball combination isn't going to be frictionless. Thus your ball's density (or more accurately, the surface-to-mass ratio) and spin will affect the results:
A dropped golf ball will hit the floor faster than a dropped ping-pong ball due to air resistance.
It takes energy to take a ball from a non-rotating state to a rolling state. That energy will come from the total PotentialEnergy of the drop. For a ball rolling down a ramp, the totalPE is converted into rotational-energy plus speed-energy. And the speed-energy is a combination of horizontal and vertical motion, with the horizontal speed also being derived from the total PE; lowering the amount of PE left available to produce vertical speed. For a dropped ball, the total PE is converted into vertical speed*only. So the dropped ball will reach the floor before the ramped ball.
Then there is the Magnus effect. Top spin on a ball (which the ramp will cause the ball to have) converts rotational-energy and velocity into downward lift through air-resistance. The ramped ball will hit the floor sooner than a ball with the same velocity* but without the rotation. On the ramp itself, the Magnus effect will have a braking effect on the ball. By choice of material of the ball (surface friction and surface-to-mass ratio) and ramp (surface friction), one can minimize the Magnus effect. However, the effects of top spin, bottom spin, and side spin are quite noticible in ping-pong, tennis, golf, baseball, soccer, etc.
Not saying that the experiments wouldn't be fun to do. Just be aware that confounding factors exist which tend to pull the physical results off of the answers calculated from a "let us start with a spherical cow" equation.
* Ignoring the effects of air resistance here solely for simplification.
If you use the same ball every time, wouldn't you end up with a good approximation of the parabolic path of it's travels? I mean, if you compare different types of balls in a classroom filled with air (especially air that's been disturbed by mathematical exercises), you'll get different answers. But each individual ball is going to give you a fair approximation of the parabolic formula Ic is looking for, isn't it?
In my method, the use of humans to click the stopwatches and shout out the "start" and "stop" signals is sort of a failsafe. The deal isn't that humans have slow reaction times, but that different humans are going to have fairly similar reaction times, so the times recorded are going to have some slop, but averaged over enough trials, it'll work out to a good number you could use.
I tried it dropping a pen from 2 ft. I could tell the difference between when I let go and when it hit the ground. I could say "start" and "stop" at the appropriate times without too much latency. I think I could hit a button on a stopwatch with even less latency.
Anyway, it's just a suggestion for a way to do this without having to rig up strobes or videotape -- assuming those are unavailable. I think those other methods are far superior if you have the equipment and can control the setting appropriately.
I think strobe is preferable, by the way. On the video, you will have frames that are too blurry to measure the position of the ball. And I wouldn't trust the pause button or frame-advance on a standard VHS playback unit to be anything like a standard increment. Could be wrong there, but I've had bad luck trying to freeze frame movies.
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posted
Quick aside: I was reading I, Robot the other day and there's a couple of cases in there where they're monitoring robots who are doing screwy things. In one of them, the two guys are actually continuously switching off 4 hour shifts watching over the course of days and I'm thinking to myself "Why the heck aren't they recording this?" It's weird how that idea seems so obvious to me, but apparently didn't occur to Uncle Ike.
posted
Galileo figured out the formula describing an object freefalling to earth with a ball&ramp while using a song as the timer. So it can be done with a stopwatch, at least if the ramp is long enough.
Just wrote to point out the complicating factors which have to be taken into consideration lest ya get results which confound.
quote:Galileo figured out the formula describing an object freefalling to earth with a ball&ramp while using a song as the timer. So it can be done with a stopwatch, at least if the ramp is long enough.
Some people seem to think that he useed a water clock, I actually thought the song theory had just come out a few years ago.
posted
The strobe is the answer! Set if for 0.05 seconds (20 hz), and you're set. Set the camera to "Bulb", etc., etc., etc.
You may have to try it with different f/stops, though.
White ball, black background with white lines.
The laser/doppler stuff is, I would think, fairly narrow-beam. You can't track the ball laterally (or vertically), you can't set a dozen of them up along your horizontal or vertical axis, and you can't rotate them to track the ball (do you want to have to do that math!)
Video would be fun, but (as Bob said) very likely too blurry. Though some higher-end cameras do have something akin to a "shutter speed" control...
Overall, sounds like fun!
Posts: 1862 | Registered: Mar 2000
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posted
I can think of a number of laser doppler dechniques that would work, using a light sheet, and so forth. But I was assuming we were looking for a low tech approach.
To add to Bob's pendulum and wet tennis ball idea, I used this in physics class using a stack of plastic blocks. We had a pendulum arm that could be pulled up against a stop, and it swung down until the arm projected a golf ball forward horizontally (the arm hit a stop and the ball kept going).
We calculated the initial velocity of the ball due to the swing. Then we calculated the height the ball would be above the ground at a certain distance, and stacked the plastic blocks up to that height, so that exactly one block would be knocked off by the golf ball. Not to brag, but I was the first to go, and I did it in one attempt. Impressed the hell out of myself.
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posted
I think the strobe and video camera ideas are great. But if you don't want to use expensive equipment, why not just set up a garden hose to shoot water in a continuous stream and have them measure the shape of the arc?
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posted
It's certainly a good visual. Calculating instantaneous velocity at any point in the stream would be tough, but it's actually pretty easy to calculate the velocity at the hose outlet.
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posted
Put a paddle wheel into the stream and watch it spin? I'm not sure how well this would work, you'd need a pretty light wheel in order not to disrupt the stream.
Yeah, that is an advantage to the strobe/video camera plan.
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)and fit equations to arrive at the formulas for this sort of motion, x = v (x) t and y = v (y) t – 16t^2, and also to be able to make predictions about how far a projectile will travel along each vector and how long it will remain in the air. They can't derive the formulas using calculus. 
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