Describe the flight of a rocket with its height above ground at time \(t\) (in minutes) given by \[s(t)=-t^3+220t^2+7500t\]

Question: when is the rocket at height zero?

Two of these solutions make sense. One does not. This helps us draw a good plot of the position function.

The velocity is the rate of change of position versus time, i.e. the derivative.

How does the velocity behave? Here is the plot.

What is characteristic of zero velocity? What is characteristic of the high point of the rocket's flight? When does this happen, and how high is the rocket? First, the time of this event.

We want the second solution (positive), and Sage starts counting from zero. And we want the “right hand side” as a decimal approximation (in minutes).

And we can now use this value of time to compute the height of the rocket at this particular moment (in meters).

We could continue to anlayze the flight further, but the above shows you the necessary tools in Sage. Instead, we plot the height, velocity and acceleration with the same horizontal scale so we can line them up and identify characteristics of the flight. First we define the acceleration with a derivative.

Now the plots: