# The Physics of Projectile Motion With a Clicky Pen

Sometimes, when I’m proctoring an exam, I end up with a little too much time on my hands. So I play with stuff—whatever I’ve got on hand. In this case, it was one of those clicky pens. It had stopped writing, so I assumed it was out of ink. Of course it might *not* be out of ink, so I took it apart to look at the ink cartridge and check. That’s when I discovered the fun stuff: If I push the empty ink cartridge down into the top of the pen, it compresses a spring. Now I can shoot the cartridge straight up. This is what it looks like (in slow motion 240 fps).

Now here’s the part I love—what kinds of questions can I answer about this shooting ink cartridge? Let’s get started.

### How High Will It Go?

Just from that video, can I figure out how high the pen cartridge gets shot into the air? Probably. Let me start with the most basic estimation. I am going to assume the launched pen travels at a constant velocity right at the beginning of the motion. In that case I can get a distance and a time to find the velocity. The bottom part of the ink cartridge moves up about 6 inches (15.24 cm) 6 frames. Since each frame is 1/240^{th} of a second, this means the starting velocity would be:

Let’s just call it a speed of 6.1 m/s (close enough). But how can we find the maximum height? Oh, sure you could just say “use one of them there kinematic equations.” Yes, that would get you the answer—but it wouldn’t be much fun. Instead, I am going to do this in two parts. First, I am going to find the time it takes the launched object to get to its highest point. I can find this using the definition of acceleration (in one dimension):

Once the ink thing leaves the spring from the pen, there is only one force acting on it—the gravitational force. This means the object will have an acceleration of -9.8 m/s^{2} in the vertical direction (free fall acceleration). The ink starts with a velocity of 6.1 m/s and ends with a velocity of 0 m/s (at the highest point). So, I know both the change in velocity and the acceleration in the above equation. This means I can solve for the time it takes the ink cartridge to get to this highest point. Putting my values in, I get a time of 0.62 seconds.

But I still need to find the highest point of this object. Let me start with two definitions of the average velocity. If the acceleration is constant (which it is in this case), then the average velocity is just the sum of the initial and final velocity divided by two (just like a normal average). The average velocity is also the change in position divided by the change in time. I can use this to find the change in height.

Boom. That’s it. I have the time (from the acceleration equation) and I have the initial velocity from the video. This gives me a maximum height of about 1.9 meters. OK, that seems reasonable.

Just for fun, I also found the launch velocity using video analysis. Here is a plot of the vertical position of the ink thingy as a function of time.

I can get the launch velocity from the slope of this line. That’s 5.84 m/s—pretty close to my first estimate.

### Checking My Answer

Really, I could just stop with my calculation. However, I am going check my answer anyway. Here is another video with the same clicky pen shooting the cartridge. In this case, you can see the whole motion (again, this is slow motion 240 frames per second).

I’m not going to get an exact measurement (that can be one of your homework questions), but it seems to go very close to 1.9 meters high. Very close for an approximation. I’m happy.

### Homework

But wait! There’s more. How about some homework questions for you? Here are somethings you can try.

**1. What is the spring constant for this clicky pen spring?** You could probably get a very rough estimate of the spring constant just by making some approximations of the ink cartridge mass and the amount the spring is compressed. However, I will also give you some more data. The mass of the ink thingy is 1.53 grams. Also, the ink cartridge moves about 1.5 cm from uncompressed to the compressed position.

**2. Suppose I want to shoot fully loaded ink cartridge, how high would it go?** You can estimate the mass of the loaded cartridge or get one and measure its mass.

**3. Here is one more video to measure the launch speed of the ink thing. Should this give a different value from before? Does it?**

Thank you for your helping hand.