A computational approach to modeling projectile motion, continued.

Here is the activity I am looking at for tomorrow in Physics. The focus is on applying the ideas of projectile motion (constant velocity model in x, constant acceleration model in y) to a numerical model, and using that model to answer a question. In my last post, I detailed how I showed my students how to use a Geogebra model to solve projectile motion.

Let me know what I’m missing, or if something offends you.

A student is at one end of a basketball court. He wants to throw a basketball into the hoop at the opposite end.

• What information do you need to model this situation using the Geogebra model? Write down [______] = on your paper for any values you need to know to solve it using the model, and Mr. Weinberg will give you any information he has.
• Find a possible model in Geogebra that works for solving this problem.
• At what minimum speed he could throw the ball in order to get the ball into the hoop?

We are going to start the process today of constructing our model for projectile motion in the absence of air resistance. We discussed the following in the last class:

• Velocity is constant in the horizontal direction. (Constant velocity model)

$latex x(t) = x_{0} + v t$

• Acceleration is constant in the vertical direction (Constant acceleration model)

$latex v(t) = v_{0} + a t$
$latex x(t)=x_{0}+v t +frac{1}{2}a t^2 $

• The magnitude of the acceleration is the acceleration due to gravity. The direction is downwards.

Consider the following situation of a ball rolling off of a 10.0 meter high platform. We are neglecting air resistance in order for our models to work.

Some questions:

• At what point will the ball’s movement follow the models we described above?
• Let’s set x=0 and y = 0 at the point at the bottom of the platform. What will be the y coordinate of the ball when the ball hits the ground? What are the components of velocity at the moment the ball becomes a projectile?
• How long do you think it will take for the ball to hit the ground? Make a guess that is too high, and a guess that is too low. Use units in your answer.
• How far do you think the ball will travel horizontally before it hits the ground? Again, make high and low guesses.

Let’s model this information in a spreadsheet. The table of values is nothing more than repeated calculations of the algebraic models from the previous page. You will construct this yourself in a bit. NBD.

• Estimate the time when the ball hits the ground. What information from the table did you use?
• Find the maximum horizontal distance the ball travels before hitting the ground.

Here are the four sets of position/velocity graphs for the above situation. I’ll let you figure out which is which. Confirm your answer from above using the graphs. Let me know if any of your numbers change after looking at the graphs.

Now I want you to recreate my template. Work to follow the guidelines for description and labels as I have in mine. All the tables should use the information in the top rows of the table to make all calculations.

Once your table is generating the values above, use your table to find the maximum height, the total time in the air, and the distance in the x-direction for a soccer ball kicked from the ground at 30° above the horizontal.

I’ll be circulating to help you get there, but I’m not giving you my spreadsheet. You can piece this together using what you know.

Next steps (not for this lesson):

• The table of values really isn’t necessary – it’s more for us to get our bearings. A single cell can hold the algebraic model and calculate position/velocity from a single value for time. Goal seek is our friend for getting better solutions here.
• With goal seek, we are really solving an equation. We can see how the equation comes from the model itself when we ask for one value under different conditions. The usefulness of the equation is that we CAN get a more exact solution and perhaps have a more general solution, but this last part is a hazy one. So far, our computer solution works for many cases.

My point is motivating the algebra as a more efficient way to solve certain kinds of problems, but not all of them. I think there needs to be more on the ‘demand’ side of choosing an algebraic approach. Tradition is not a satisfying reason to choose one, though there are many – providing a need for algebra, and then feeding that need seems more natural than starting from algebra for a more arbitrary reason.