Holiday Physics Homework

Rhett Allain

It’s that time of year. Christmas, New Years, and other holiday stuff. Now you can just sit around in your pajamas and watch football. Or can you? Don’t forget your holiday physics homework.

Before I post your homework, I am going to give you an example. If I don’t give an example, everyone gets angry. So here it is (yes, this is an old problem with new data).

How Fast Is That Nerf Dart?

Everyone loves Nerf guns. Also, if you borrow your wife’s phone you can easily make a slow motion video.

Slow motion video at 240 fps.

This is a 240 fps video from the iPhone 6 (here is a youtube version). It’s a little difficult to see, but there is also a meterstick on the ground for scaling. You should know what comes next – video analysis with Tracker Video Analysis. I’ll just skip all the details (but here is a quick video analysis tutorial) and go straight to a plot of the position of the dart.

Since the dart was shot at an angle, I have included plots for both the x- and y-position. Looking at the slope, this gives an x-velocity of 19.2 m/s and a y-velocity of 3.04 m/s. This gives an average magnitude of 19.44 m/s. This speed is quite a bit faster than this other Nerf gun I tested at around 10 m/s.

See. That’s not so difficult of a homework problem. Actually, you could take this one step farther. Here are some follow up questions.

• How much does air resistance slow the dart down? Make an estimate based on the size and mass of the dart. If the dart was shot in a space station (so you could ignore gravitational effects), how far would it travel? What is the effective range of the dart?


• Projectile motion. If you neglect air resistance, how high will the dart travel if you shoot it straight up? How far horizontally will the dart travel if you shoot it from a standing position and aimed horizontally?


• Estimate the stretch distance and the spring constant for the spring inside the gun.

Some of those questions aren’t too difficult.

Other Holiday Homework Questions

Now that you have an example, here are some other questions.

1. Twist ties. For the past few years, it seems like many toys have been packaged with an increasing number of twist ties to hold them into the package.

Experimentally create a model for the retaining force strength as a function of twists. So, if you twist it twice instead of once, is it twice as strong?

2. Twist ties and time. Estimate the total amount of time (in human hours) that was spent in Christmas morning in the USA undoing twist ties.

3. Air drag and power. Here is a running parachute.

Running with a parachute.

Estimate the increase in power needed to run with the parachute.

4. Wrapping paper. Suppose you took all the wrapping paper used in the USA during Christmas. If you put all of this together (tape it together), how much surface area would it cover? Would it be enough to cover a city? What about Rhode Island? Could you see it from orbit?

5. Hot Christmas trees. Suppose you have normal incandescent bulbs on your Christmas tree. What perecent of the electrical power goes into thermal power? Does your tree actually get hot?

Thermal image of a Christmas tree.

This older post on the power used by Christmas lights might be helpful.

That’s your homework.