Rockets and Forces
Remember the first day of class when we said that this was a course about Physics and rockets? Let's see if we can't tie some of what we have been
doing to our rocket-based course so we can get these puppies off the ground! In order to do this, let's start with a basic rocket and put it into flight in one of 4 phases: Launch, thrust,
cruise, and recovery. While doing this, let's keep Newton's 3 laws in mind:
- An object in motion stays in motion at a constant velocity and an object at rest stays at rest unless acted upon by a net external force.
- The acceleration of an object is directly proportional to the net external force acting upon the object and inversely proportional to its mass.
- When an object applies a force on another object, the second object applies a force to the first object, equal in magnitude but opposite in direction to the force applied by the first object.
The Launch Phase (for the purposes of this class only) will establish the initial conditions for our flights. This isn't really one of the phases of flight, but it is helpful for establishing our initial conditions. By this, I mean that the rocket is sitting on the launch pad,hooked up to our ignition system, and ready to go. The engine is not firing. Our free body diagram looks like:
There are only two forces acting on the rocket - the weight of the rocket down (i.e.the force on the rocket due to gravity, Fg or mg) and the Normal Force (the force of the launch pad pushing back up on the rocket, FN). We've been doing these problems long enough to know that, since the rocket isn't moving, the Normal Force is equal to the weight of the rocket. Other initial conditions:
Initial Velocity (Vo) = 0 m/s
Mass of the Rocket is the take-off mass and includes the full mass of the engine propellant since none has burned off yet. This will vary for each rocket, but a typical value for mass is .05 kg. A typical small engine may have .0035 kg of propellant.
The Thrust Phase is the point in the
flight from the time the rocket engine fires until the rocket engine stops firing. Typically, this can last from 0.2 sec for the smallest engines we will use to 2.1 seconds. Since we will be
manufactured engines, we can predict this number fairly accurately using the manufacturer's design data. During this time, there are 3 forces that will be acting on the rocket: The weight of the rocket (we just can't escape gravity in this course!), the thrust provided by the engine, and the drag or the force due to air resistance that will oppose the motion of our rocket and cause our predicted altitudes to be really off. We'll discuss the drag forces in some detail later. But our free body diagram now looks like the example to the right.
If we take a moment apply Newton's Second Law, then we see
From here, we can do a couple of different things: We can ignore drag and figure out the average acceleration on the rocket in this phase of the
flight in order to determine the final velocity at the end of the phase, or we can approximate a value for drag and do the same thing. Remember, our mass is changing because the propellant is
burning out during the flight. So we'll also use an average mass during this time frame.
The Cruise Phase occurs from the time that the engine burns out until the recovery device is ejected by the engine. Time delays built into the engine will allow us to select how long this phase lasts. During this phase, the rocket will fly to its highest point (apogee) after which we hope that the parachute will pop out. Only two forces act on the rocket now: Weight and drag. The free body diagram is relatively simple (to the right) and our net force equation looks like:
Again, we can make an assumption that air resistance is zero and solve for our expected maximum altitude using simple kinematics equations - the rocket reaches its maximum velocity at the end of the thrust phase and is now slowing down due to gravity until the velocity of the rocket is 0 and the rocket is at its highest point. Drag on the rocket will cause that altitude to be much lower than calculated without drag. Hopefully, we will select our engines such that the parachute does not deploy until after we reach our highest point.
Our final phase, the Recovery Phase, occurs after the parachute deploys. At this point, the rocket is drifting (we hope) gently back to the ground.
As we can see on our free body diagram to the left, there are now 3 forces acting on the rocket - gravity, the resistance provided by the parachute, and the ever-present air resistance. Note
that the drag vector has switched direction - it is opposing the downward motion of the rocket and actually helps our recovery effort. Our net force equation looks like
Fparachute +Fdrag -mg = ma
and we can use this to predict the velocity that the rocket will have when it lands. Parachute drag is a function of the size and shape of the parachute and we'll look at some models for that in a later lesson.
There is one more force that we have totally ignored here. We have assumed that our rocket is going to go straight up and come straight back down. In reality, it is very likely that we will have a breeze when we are flying and so we will need to deal with the effects of the wind on the rocket. We will simplify this somewhat by making the assumption that the forces discussed above apply in the vertical, or Y-axis and that wind will only have an effect in the horizontal, or X-axis.