# More projectile motion than you can shake a stick at

I don’t really know what that title actually means. So, I have been having problems with my PASCO projectile launcher devices. I will just call them launchers (they are really cannons). In my previous post, I looked at the launch speed from a launcher shot horizontally and vertically. The problem was that I was getting different launch speeds for the vertical and horizontal shot. So, here is my plan: shoot the ball and a variety of angles from 0 to 90 degrees and see how the launch speed changes. I will only use the data from video analysis (of course using Tracker Video Analysis)

PASCO makes two launchers. A normal sized one and a mini-me-launcher.

So, what data do I actually get from the video?

• time
• horizontal (x) position
• vertical (y) position
• I measured the angle from the launcher

Also, after getting the data, Tracker Video fit a function for me so that I have:

• The slope of the x-t graph (this is the x-velocity)
• The parameters of the 2nd order polynomial that fits the y-t data
• The time of the first data point. This is not zero for two reasons. First, the video frame was likely taken some moment after the ball left. Second, I didn’t cut the video up for each shot. So, the 30 degree shot may have taken place at 15 seconds into the video.

So, I have two ways to get the launch velocity.

### Launch Velocity Method 1

The x-velocity is pretty straight forward to get. Since the x-velocity is constant, I can just find it from the slope of the x-position time graph. Actually, this is a good measure to see if things are working correctly (a straight x-t graph).

With the angle and the x-velocity, I can find the total velocity when the ball was launched.

A couple of problems. What happens when I shoot the ball straight up? Then the x-component of the velocity is zero. So, it doesn’t work for this. Also, the higher the ball is shot, the less accurate this will be. Also, this assumes I have the launcher at the correct angle. I used the Tracker Video angle measuring tool and it essentially agreed with my measurement.

### Launch Velocity Method 2

This next method finds the total launch speed from both the x- and y-velocites using:

How do I find the initial y-velocity? I suspect this is where my error from my previous post came. If I wait until the next video frame, I will not be finding the initial velocity. Well, when I fit a polynomial to the y-time data, I get:

NOTE: That a is a parameter, it is not the acceleration. If I take the derivate with respect to time, I get:

That would be great if I knew the time that I want to call “initial” velocity, but I don’t. I can find the position of the ball initially though. In fact, when I took data I chose the launch position as the origin. In this way, when the ball is at x = 0, y = 0, that is the velocity I want. Someone please tell me why I only wrote down the polynomial fit for the y-direction and not the x? That was dumb. Anyway, I have the function for y(t). I can find the time that corresponds to y = 0 meters and then use this time for the velocity as a function of time.

This doesn’t really work well when theta is 90 degrees. For this case, I plotted y vs. vy and I get something like:

This gives me y as a function of vy. I set the origin to y = 0 so that I can solve for the “roots” of this equation. The positive root will give me the y-velocity at y=0.

### Data

You have been very patient. Here is what I found. First, for the small launcher.

The red line is the data from calculating the launch speed only from the angle and the x-velocity. This one does not have a value for theta = 90 degrees, you know…because. Also, to get a sense of how the function fits varied, here is a plot of the acceleration of the ball for the different shots.

The acceleration for all of these seems off of the accepted -9.81 m/s2 value. That could easily be due to a scaling problem. The acceleration for the theta = 0 degrees shot is probably off because there were very few data points in that case.

Here is the similar data for the large launcher:

I don’t think the speed of the launcher depends significantly on the angle. First, if it did, I would expect to see some type of trend for v0 vs angle. I don’t see a trend. Second, I don’t suspect that the acceleration of the ball changed with each shot, yet it produced a similar looking plot.

I must have just made some type of mistake in my previous attempt. So, I shall certify the PASCO launcher as being an acceptable device. Oh – and I really should have error bars on those graphs, but I have not decided the best way to get the uncertainty from a video analysis. I am still thinking about that one.

1. #1 JimP
October 26, 2009

I would love your thoughts on uncertainty in video analysis? What even does multiple trials look like? Do I take several videos if like this one it is an experiment I can repeat? Do I track the object multiple times in the same video? What about setting the scale multiple times?

2. #2 Rhett
October 26, 2009

@JimP,

You are correct – I have neglected the issue of uncertainty in video analysis for too long. This will have to be a blog post soon. But, I think the best way to treat it is to not do multiple runs (although that might be the easiest in the end) – rather think of each point has having an uncertainty. Let me think about this and write it up.

3. #3 Steve Huling
December 4, 2009

I have used the large Pasco launcher for years in my high school physics class and I’ve noticed discrepancies with the launch velocity as the angle changed. Pasco sells a bracket that allows you to put 2 photogates just past the barrel and these photogates are 10.0 cm apart. Timing the ball as it comes out the barrel gives you a pretty direct method of measuring launch velocity. I decided to measure velocities at 20 degree intervals and the results made no sense whatsoever. On low power, the velocity seemed to decrease as I went from horizontal to vertical (4.43 m/s at horizontal, 4.20 m/s at vertical) while on high power, the velocities actually showed an increase (9.26 m/s to 9.61 m/s). Medium power didn’t show any difference! Weird. I had contemplated doing video analysis but then just decided to to do some 30 and 60 degree shots, measure the horizontal distance traveled (balls landed at same height as the barrel) and calculated the launch velocity from that. The results were completely different, I actually got very similar launch velocities for a 30 degree shot as a 60 degree shot for low, medium and high “power”. Why wouldn’t photogates give comparable velocities? I know the ball doesn’t travel in a straight 10.0 cm line as it leaves the barrel and passes through the 2 photogates but I’m still surprised at the difference in velocities.

Here is an example: My newer launcher gave me velocities on medium power gave me a velocity of 6.00 m/s at a 30 degree angle using photogates and when I fired it at a 30 degree angle and calculated velocity based on range, I got a value of 6.49 m/s. Not even close!