## Parameters

Trajectory is a motion modeling program that allows a user to explore the effect of a variety of parameters on the two-dimensional motion of an air-borne object. In order to effectively model the motion of an object, we need to know some information about the object. We're not asking for names, addresses or social security numbers. But how about the initial height, speed, launch angle, the mass, profile area and other geeky parameters that you would only dare share with your good friends at The Physics Classroom? Exactly what we are asking for is described in detail below. It might require a bit of thought, some research, and some strategic planning. But when you hand over the info, we can model the motion and help you out with your project.

**Axis Convention**: No decision to be made here ... but something you need to know. Our program defines + direction as being upward and the - directioni as being downward. Thus, a negative air resistance force indicates a downward air resistance.

**∆time (s)**: We perform calculations of all motion quantities on a periodic time interval, defined as ∆time. The value you enter is the time increment at which calculations are performed. Smaller values mean greater accuracy but more data. Here's a general guideline: A large value (0.05 and greater) results in inaccuracies (especially for low initial heights). A small value (0.01 and smaller) results in excessive amounts of data (especially for tall initial heights). For initial heights greater than 1000 m, a value of 0.1 works well. For initial heights less than 50 m, a value of 0.01 or smaller is likely needed.

**Init. Height (m)**: The initial height indicates the distance above the ground when t = 0.0 seconds. The object must start with a positive height (above the ground). But don't feel a need to start at 0.0 m? Let loose. Get wild. Dream big. We can handle several thousand meter heights quite easily. We're not afraid of height; you shouldn't be either ... at least not with the Trajectory program. User tip for the non-metric among us: 1.00 meter is equivalent to 3.28 feet.

**Init. Velocity (m/s)**: This is the speed of the object at t = 0.0 seconds. We'll take any value you give us as long as it's not a negative value.

**Launch Angle (°)**: The object can be dropped from rest, projected upward, or projected downward. This information is conveyed by the launch angle. It must be a value between -90° and +90°. A value of 0° indicates a horizontal launch (from an elevated position). A value 30° indicates a launch that is 30° above the horizontal. And finally, a launch angle of -30° indicates the object is initially moving in a direction that is 30° below the horizontal.

**g (N/kg)**: The symbol **g** stands for the gravitational field constant. It is sometimes called the acceleration of gravity or the acceleration caused by gravity. Since gravity acts downwards, you must enter a negative value. We'll shame you if you don't. Since it is a variable, you can use 9.8 N/kg for Earth or even run an experiment on the moon (or your favorite planet) where the value of **g** is different. Field trip, anyone?

**Mass (kg)**: Identify the mass of your object. Don't be bashful, we won't tell anyone. For those who aren't too *chummy* with the unit kilogram, consider this: A 100-pound object on Earth has a mass of 45.5 kg. Now you have a conversion factor to find the mass of about any object you wish to experiment with. The minimum mass our program allows is 0.20 kg. Because of some assumptions behind the calculations, it does not model light-weight objects (coffee filters, feathers, leaves) accurately.

**Profile Area (m ^{2})**: This is a difficulty concept. You will need to understand the concept so that you can estimate the area intelligently. Air resistance acts against an object's motion. It's a "head-on force" so you need to know how much area of the object is moving head-on towards the air. That's what the term profile area tells us. Technically, it's the area of the projection of an object on a plane that is perpendicular to the object's direction of motion. So you have to think ... if you stood on the ground and looked up at the object from directly underneath, what would the area look like or look most like? Does it look like a circle or a square or a long, thin rectangle or a ...? So it doesn't look like any of these? What could you estimate it as? For the person falling in a spread eagle position, it looks most like a rectangle to me. Now you need to know the dimensions of the object. Consider pulling out a meter stick and using it as a reference for estimating a dimension or two - like the radius of a circular area or the length and width of a rectangle. Use an area calculation to determine the area in m

^{2}.

**Drag Coefficient**: This unit-less coefficient provides a measure of how efficiently the air streams around the object. Use the Wikipedia page to assist in obtaining an estimated value for your object. Attempt to match the shape of the object you're studying to one of the shapes on the page. Find the drag coefficient for the most approximate shape you find. Or take the average of two coefficients based on the closeness of your object's shape to the two shapes shown.

**Air Density (kg/L)**: The density of air through which the object is falling impacts the amount of air drag encountered by the object. Air density depends on several factors. Our program assumes this value to be constant despite the fact that it is probably changing over the course of the fall (if this distance traveled is a large distance). Use the Wikipedia page to provide an estimated value. If part of your study involves a comparison of falling with air resistance to falling without air resistance, you can enter 0 for the air density to model falling without air resistance.

**How to Break the Program**

We're not going to lie. You can break the program. Hopefully you have to do something rather odd to break the program. For instance, if you set up a trial in which you throw an object downward with an initial velocity of 300 m/s from an initial height of 2.0 m, you're probably going to break the program. (You're also probably going to break the object too but we won't have any data to back that up.) The worse thing that happens when you break the program is you have to reload the browser window (or tab). It's not a big deal. The bigger deal is you might want to try to model something rather odd and our program doesn't do it. It's set (we hope) to model realistic scenarios ... and not the type of scenarios you might see on your favorite cartoon or action figure show.

The most important way to avoid breaking the program is to consider how it works. It calculates values of height, velocity, air resistance, net force, and acceleration every interval of ∆time. It then checks to see if the object has hit the ground or moved below ground, at which point it stops calculating and displays the data. So the value of ∆time relative to your other inputted values is critical. If an object is thrown up or down at 300 m/s (unrealistic), then it is important that our program perform frequent calculations - like every 0.001 seconds. The closer the initial height is to the ground, the more important it is that the ∆time is small so that we can detect the ground collision before the object collides with Earth's magma. On the other hand, an object that is released from a tall initial height (like 5000 m) can have a larger ∆time. Since it will be falling for quite a while, you can lessen the load on the program by repeating calculations every 0.1 second. When the height is tall and the ∆time is small and many calculations are done, there may be a small delay before the data are displayed since the program is still busy with its calculations. The program is not broken, it's just busy calculating. Feel its pain and change the ∆time the next time you do a trial.