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 2^nd 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.

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.

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. v_y 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.

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.