The Big Idea: If you shot a bullet perfectly horizontally, and at the exact time that the bullet exited the end of the barrel you dropped another bullet from the same height, which bullet would hit the ground first? Answer: They would hit at the same time! I'll prove it...
Summary: By the end of this instructable you will be able to teach your students the ins and outs of two-dimensional trajectory problems in a fun and engaging manner. This project allows a shooter to hit a falling stuffed toy every time without missing, like magic! Additionally, you will be able to explain why this happens and your students will be able to solve all of the math themselves. You'll have to excuse the lack of set up pictures as I do not have a classroom available for the set up at this time.
* Two-dimensional trajectory
* Basic equations of motion (position, velocity, and acceleration)
* Two body collisions
* Directional forces and acceleration
This instructable is very versatile. I will present alternatives to my options as we go along. However, here are the materials I used:
1) 4 feet of 1/2 inch diameter PVC piping
2) 2x Chemistry ring stand
3) 2x Burette clamp
4) 1x C-cell battery holder (4 batteries)
5) 1x Marble
6) 1x Electromagnet --> See link on the circuit building page for the one I use.
7) 1x Stuffed toy (I use a stuffed monkey)
8) Spool of small gauge wire
Now before we start we must develop the physics and math behind our project...
Step 1: Trajectory Theory
The math and the physics behind this project are fairly simple. First I will develop all the math and theory you need to know, in its most basic form, to teach this lesson (I'm assuming you know basic physics). Then I will go into some more complicated options for advanced students. I recommend briefly reading this section, viewing the rest of the instructable, and then returning to this page once you have a solid idea of what is happening in the experiment.
There are two bodies in motion in this project. The first is the bullet (the marble) and the second is the monkey (or other stuffed toy). A volunteer shooter stands on one side of the room and shoots the marble out of the end of the PVC tube like a spitball. On the other end of the room hangs a stuffed toy from an electromagnet. As the marble exits the end of the tube, it breaks a small wire contact. When the wire contact is broken, the electromagnet shuts off, dropping the toy. The marble will hit the falling toy every single time, no matter the speed of the ball bearing. Students are always amazed by this.
The Basic Theory:
(Forgive me for the length of this explanation) In the 2D plane there are obviously 2 directions: the x and the y. Therefore, in order to accurately describe a body's position and motion in the plane, you must describe its x position, velocity, and acceleration, as well as its y position, velocity, and acceleration. Since this experiment does not occur when the bodies are in equilibrium, it will be necessary to also describe the body's position, velocity, and acceleration at different times. Don't worry, it's not that complicated just take it steady.
Equations of Motion:
If you are familiar with physics you should know the equations of motion (with constant linear acceleration):
Position: s(t) = s0 + v0*t + 1/2*a*t^2 , where s = position (x or y), v = velocity (x or y component), a = acceleration (x or y component)
Velocity: v(t) = v0 + a*t
acceleration: This will be either 0 for horizontal directions or the acceleration due to gravity in vertical directions.
Using the Equations:
We have two different bodies so we'll have to look at each one separately.
Body 1, the marble:
The marble's "initial state" will be the state it is in when it leaves the end of the barrel. Therefore, it will have only an x component of initial velocity. If we neglect air resistance and friction, it will not have an x component of accleration. So, our x-component of position for the marble looks like this: x(t) = x0 + v0*t. If you assign your coordinate system such that the initial x-position of the marble is zero (as in my figure), the equation is simply x(t) = v0*t. You can use a photogate at the end of the barrel to get the bullet's exact velocity, or you can make it up for the sake of illustration.
Since we want to know the time and position the two bodies will collide, and since the monkey is dropping straight down (no change in x-position) that means we want to know the time at which the x-position of the marble is equal to the x-position of the monkey. You'll have to measure this distance when you set it up in your classroom. Let's say you measured it to be 25 feet and that the ball bearing is moving at 50 feet per second. This is what the equation would look like: 25 ft = 50 ft/s * t. Solve for t to get that the marble will be at the same x coordinate as the monkey at t = 0.5 seconds.
Body 2, the falling monkey:
We now know when the marble will arrive and hit the monkey. But we need to find out where the two will be in vertical space when they collide. Take the y-position equation for the monkey: y(t) = y0 + v0*t + 1/2*a*t^2. Now in this case, y0 will be initial height that the monkey is hung at, it will not have an initial velocity, and it's acceleration is the acceleration due to gravity. So, the equation looks like this: y(t) = y0 + 1/2*(-32.2 ft/s^2) * t^2. Note that if you did the y-position for the marble you would come up with the exact same equation. This means that they will always be at the same y-position!!!
Ok, so we found out they will hit at t = 0.5 seconds so let's plug that in (assume the initial height of the monkey is 6 feet off the floor):
y(t) = 6 ft + 1/2 * (-32.2 ft/s^2) * (.5 sec)^2. Solving for the y-position we get y = 1.975 feet. Ta daaa! The bullet will hit the monkey after half a second and while they are 1.975 feet above the floor.
For the advanced students / educators:
You can make this as complicated as you would like. Have the students derive the equations, or propose a scenario with non-constant accelerations and velocities (note: this no longer guarantees they will hit each other). If they know calculus, help them realize that the equations of motion are derivatives of each other.