Making use of Mathematica in Physics

It's never easy to know how much technology to use when teaching science and math. I'm coming over to the philosophy that an appropriate aspect of the process of thinking about that is to factor in what I do myself, when I'm working the problems sets for myself. I don't balance my check book without reaching in the drawer and getting out a calculator. If I have "modeling" to do for personal finances, I'll use a spreadsheet. If I'm working a physics problem set, I'll often turn the mathematics over to a piece of software called Mathematica.

When I'm trying to explain the physics to a student, I want to focus on the physics concepts, and not spend a lot of time going though algebra steps or graphing steps the student already knows, but we all know can take quite a bit of time. Mathematica allows me to use my time to focus on the physics.

Some Basics of Mathematica

To solve an equation, we use the command Solve, the syntax of which is

Solve[the equation written with a double equal sign, the variable to solve for]

For example the equation for the time it takes an object dropped from rest to fall 100 meters, under the influence of gravity would be:

The work above is the physics, solving for t is the mathematics, and that we can turn over to Mathematica by using the Solve command.

Here is this equation solved in Mathematica, what you input is shown in the line called In[1], the solution you get back is shown in the line called Out[1].

And here is a quick and easy plot of the distance an object falls under the influence of gravity in time t.

The command is Plot. One way to handle the syntax is to define the function using an underscore after the variable name on one line, and then use the Plot command in the next line. The syntax of the Plot command is:

Plot[ name of the function to plot, the domain you want it plotted over]

I did leave off the labels for the axis to make the syntax easier and to save time.

With a graph quickly in place, now you can talk about the physics that is going on in the graph. Does the object fall the same distance during each one second interval? And then on to another question about the physics, does the velocity change by the same amount during each one second interval?

Recall that velocity is the derivative of the position, so let's make a graph.

And what is happening to the acceleration, which is the second derivative of the position?

Or maybe you have a more complicated problem where you want to focus on conservation of momentum and not have the focus be on the mathematics. Two objects of mass m1 and m2, each 10 kg collide. The initial velocity of m1 was 20 m/s, and m2 was sitting at rest. After the collision m2 is observed to leave at an angle of 30 degrees above the horizontal with a velocity of 6 m/s. What will the velocity and angle of departure for m1 be after the collision.

From the conservation of momentum, the physics is:

Turning the algebra over to Mathematica looks like this:

I'd be interested in hearing if other physics teachers agree or disagree with these thoughts.