Modeling Scenario


Author(s): Brian Winkel

SIMIODE - Systemic Initiative for Modeling Investigations and Opportunities with Differential Equations

Keywords: resistance Flight Rocketry mass Newton's Second Law of Motion thrust Free Body Diagram apex rocket

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Resource Image We offer an opportunity to build a mathematical model using Newton's Second Law of Motion and a Free Body Diagram to analyze the forces acting on the rocket of changing mass in its upward flight under power and then without power followed by its fall to


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Article Context


Suppose we have a small rocket with a full 100 liter fuel tank. Fuel (with a mass density of 0.98 kg/l) is burned at a steady rate of 3 liters per second and can provide a constant thrust force of 5900 Newtons as it burns. The rocket and fuel tank has a mass of 400 kg with no fuel in it. The shape design of the rocket causes a resistance proportional to its velocity during flight of 2 v(t) where v(t) is in m/s and the constant 2 for proportionality of resistance is 2 N/(m/s). We aim the rocket straight up and launch it . . . 5  . . . 4 . . . 3 . . . 2 . . . 1 . . . Launch . . . We have lift off!!!

How long will the rocket burn and hence provide thrust? We call this time interval the burn period.

Build a differential equation model for the powered flight (during the burn period) of the rocket based on all the External Forces and Newton's Second Law of Motion.

Article Files


Author(s): Brian Winkel

SIMIODE - Systemic Initiative for Modeling Investigations and Opportunities with Differential Equations



  1. 0 Like

    Hadas Ritz @ on

    This is a great example problem. I used it during lecture today for an example of generating a 1st order model. I mostly worked through the problem myself, but I used some polling questions about what forces to include, which forces were and weren't constant. I also had the students think about and discuss how to find the burn time (some used dimensional analysis, some made a differential equation).

    During class a student pointed out one error: the force sum should equal $$\dot{m}v+m\dot{v}$$ because both mass and velocity are changing with time. I had written just $$m\dot{v}$$, as the posted activity Teacher version also does. The good news is that this doesn't stop it from being a linear equation! (Just a little more complicated.)

    When looking carefully at the posted files, I also discovered a second minor error: because this is a linear drag model, and not a quadratic drag model, you don't actually need a new FBD nor a new model equation during the down-ward velocity portion of the flight. The drag force "automatically" changes direction when the velocity does. 

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