The Physics of Projectile Motion With a Clicky Pen



Typically, after I’m proctoring an examination, I find yourself with a little bit an excessive amount of time on my fingers. So I play with stuff—no matter I’ve bought readily available. On this case, it was a type of clicky pens. It had stopped writing, so I assumed it was out of ink. After all it’d not be out of ink, so I took it aside to take a look at the ink cartridge and examine. That is after I found the enjoyable stuff: If I push the empty ink cartridge down into the highest of the pen, it compresses a spring. Now I can shoot the cartridge straight up. That is what it seems like (in sluggish movement 240 fps).

Now here is the half I really like—what sorts of questions can I reply about this capturing ink cartridge? Let’s get began.

How Excessive Will It Go?

Simply from that video, can I work out how excessive the pen cartridge will get shot into the air? In all probability. Let me begin with essentially the most fundamental estimation. I’m going to imagine the launched pen travels at a relentless velocity proper at first of the movement. In that case I can get a distance and a time to seek out the speed. The underside a part of the ink cartridge strikes up about 6 inches (15.24 cm) 6 frames. Since every body is 1/240th of a second, this implies the beginning velocity can be:

Let’s simply name it a pace of 6.1 m/s (shut sufficient). However how can we discover the utmost top? Oh, certain you can simply say “use one among them there kinematic equations.” Sure, that might get you the reply—but it surely would not be a lot enjoyable. As a substitute, I’m going to do that in two components. First, I’m going to seek out the time it takes the launched object to get to its highest level. I can discover this utilizing the definition of acceleration (in a single dimension):

As soon as the ink factor leaves the spring from the pen, there is just one power appearing on it—the gravitational power. This implies the article could have an acceleration of -9.eight m/s2 within the vertical path (free fall acceleration). The ink begins with a velocity of 6.1 m/s and ends with a velocity of zero m/s (on the highest level). So, I do know each the change in velocity and the acceleration within the above equation. This implies I can resolve for the time it takes the ink cartridge to get to this highest level. Placing my values in, I get a time of zero.62 seconds.

However I nonetheless want to seek out the best level of this object. Let me begin with two definitions of the typical velocity. If the acceleration is fixed (which it’s on this case), then the typical velocity is simply the sum of the preliminary and remaining velocity divided by two (similar to a standard common). The common velocity can be the change in place divided by the change in time. I can use this to seek out the change in top.

Growth. That is it. I’ve the time (from the acceleration equation) and I’ve the preliminary velocity from the video. This provides me a most top of about 1.9 meters. OK, that appears cheap.

Only for enjoyable, I additionally discovered the launch velocity utilizing video evaluation. Here’s a plot of the vertical place of the ink thingy as a perform of time.

I can get the launch velocity from the slope of this line. That is 5.84 m/s—fairly near my first estimate.

Checking My Reply

Actually, I may simply cease with my calculation. Nevertheless, I’m going examine my reply anyway. Right here is one other video with the identical clicky pen capturing the cartridge. On this case, you’ll be able to see the entire movement (once more, that is sluggish movement 240 frames per second).


I am not going to get a precise measurement (that may be one among your homework questions), but it surely appears to go very near 1.9 meters excessive. Very shut for an approximation. I am glad.


However wait! There’s extra. How about some homework questions for you? Listed here are somethings you’ll be able to strive.

1. What’s the spring fixed for this clicky pen spring? You can in all probability get a really tough estimate of the spring fixed simply by making some approximations of the ink cartridge mass and the quantity the spring is compressed. Nevertheless, I will even provide you with some extra information. The mass of the ink thingy is 1.53 grams. Additionally, the ink cartridge strikes about 1.5 cm from uncompressed to the compressed place.

2. Suppose I need to shoot absolutely loaded ink cartridge, how excessive would it not go? You possibly can estimate the mass of the loaded cartridge or get one and measure its mass.

three. Right here is yet another video to measure the launch pace of the ink factor. Ought to this give a distinct worth from earlier than? Does it?


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