This is really a lab that I have students do, but I am pretty sure they don’t read this blog – so it is ok. If they are reading this, hi!

We have these projectile cannons that shoot small balls. In order to look at projectile motion, they need to first determine the launch speed of the ball. I have a great method for this. Basically, shoot the ball horizontally off the table and measure how far horizontally it goes. You can get the final location of the ball by having it hit a piece of carbon paper on top of normal paper. If you don’t know what carbon paper is, you are young.

Anyway, after doing this lab for a couple of semesters, I noticed that sometimes students would not read the instructions (I know, it’s shocking, but true). Instead of using the vertical distance the ball falls to get the time, they used a stop watch. So, this year I changed the lab (I think I also got a suggestion from some blog somewhere). Actually, projectile motion is now two labs. In the first lab, the goal is to measure the launch speed (with uncertainty) and then the second lab looks at projectile motion. I have the students find the launch speed several ways and compare uncertainties for the different methods.

- Method 1: Launch the ball straight up and measure the height.
- Method 2: Launch the ball straight up and measure the time of flight.
- Method 3: Launch the ball horizontally off the table and measure the vertical and horizontal distance.
- Method 4: Launch the ball horizontally and measure horizontal distance and time.

### Uncertainty

First, this is not real uncertainty. This is cheating uncertainty. The basic idea is that students calculate the maximum and minimum values a quantity can be and use that for the uncertainty. More details here – with an example.

### Method 1

Here you would measure just the height the ball goes (and you would assume the ball accelerates in the negative y-direction at 9.8 m/s^{2}). To get the initial velocity, I will say that the average velocity is (in the y-direction):

In case it wasn’t clear, the final velocity was zero m/s. I can say this because the velocity is changing at a constant rate. Also, I can write down the definition of average acceleration (in the y-direction):

Finally, using this and the definition of average velocity (a different definition) (again in the y-direction):

You could also get this using the work-energy principle, but there it is. If I assume that there is no uncertainty in g, then here is a calculation of the velocity and the uncertainty in the velocity. NOTE: To get the uncertainty in the height, you could just shoot the ball once and then estimate the uncertainty in the height. OR…you could do it like 5 times and find the standard error.

I did not round the numbers to the correct decimal place because I don’t know how to make zoho sheets do that.

### Method 2

This is similar to method 1 except that I will measure the time to go up and back down. There is a trick here. If the acceleration is constant, then the speed of the object when it leaves the cannon is the same magnitude that has when it comes back to that level. So, starting with the definition of average acceleration (in the y-direction):

For this case, I am going to measure the time interval 5 times to determine the uncertainty in time.

I changed my mind. Initially, I was just going to use the standard error for the uncertainty in time. However, I felt it was too low (which could be due to systematic error). Really, my reflexes aren’t that good.

### Method 3

This is two-dimensional motion. The key to 2-d motion is that the horizontal and vertical motions can be treated independently except they have the same time. The acceleration in the x-direction (horizontal) is zero and the acceleration in the y-direction is -g. First, looking at the y-direction the initial velocity is zero so that:

Now I can use this to solve for the time interval:

For the x-direction, I have the simple equation:

And using the above expression for the time interval I get:

Remember that the velocity in the x-direction does not change (so it doesn’t matter if you call it v_{1} or just v). Also, since the ball was shot horizontally, then the initial velocity (total) is the velocity in the x-direction.

### Method 4

This is probably the most straight forward method (perhaps why students like it). Instead of measuring the height, I will measure the time. Then I can calculate the velocity in the x-direction as (which is the total initial velocity):

Simple.

### Note

I did not look at this – but it is possible that the cannon has some variability in its firing. You could explore this if you shot it several times and see how the distance changes.

### Conclusion

Using my crude estimates, here is what I have for the 4 methods:

- Method 1: v = 2.90 +/- 0.03 m/s
- Method 2: v = 3.0 +/- 0.5 m/s
- Method 3: v = 1.80 +/-0.03 m/s
- Method 4: v = 1.6 +/- 0.4 m/s

Odd that the upward firing velocities are so much different than the horizontal firing ones. Hmmmm…. Well, method 1 and 3 have the lowest uncertainties. I think my estimation for the height in method 1 was a complete guess. Really, I should take more data, but the point was to show how to calculate the uncertainties and the initial velocities. Did that.