Learning Objective |
In this lesson the student will use mathematics to determine the velocity of a rocket. By the end of this lesson the student should have an appreciation for the real world use of mathematics as it relates to model rocketry. |

Grade Level |
9 – 12 |

How fast does a model rocket go? This may be the second most asked question after witnessing a model rocket launch. Unlike a cannon ball shot from a cannon the velocity of the model rocket actually increases after it leaves the launch pad. In this lesson we will determine the maximum velocity of a model rocket.

#### Using Data from ThrustCurve.org

There is an organization that collects data on the performance of rocket motors called ThrustCurve.org. Figure 1 is an image taken from the ThrustCurve.org website showing the thrust curve of an Estes B6 type motor (it is important to note that the manufacturer must be taken into account as the thrust curves differ depending on the manufacturer).

It is not necessary to take the delay, or the last number in a motor classification into account when we are looking for the thrust curve of our motor. We are only concerned with the thrust phase. Thus if you search for the thrust curve for a B6-4 motor it would be the same as the thrust curve for a B6-6 motor from the same manufacturer.

The first value we will observe from the thrust curve is the Average Thrust. For this motor the Average Thrust is 5.2 Newtons. Of note here is the rounding up of this value in the motor classification (B6).

The second value to observe is the burn time. As we can see from the graph it is 0.85 seconds.

#### How fast should my model rocket go?

We may determine the velocity using the following formula:

You may notice that the unit newtons is used for both the Force of the motor and the Weight of the rocket. We can measure the mass of the rocket (plus engine) and find its weight. To do so we must convert the mass in grams to weight (due to gravity) in newtons. Below is the formula for this conversion:

The mass (m) is measured in kg and the g represents the gravitational constant measured in m/s^{2}. On the Earth the gravitational constant is 9.8m/s^{2}. Returning to our original forumla we can see that 1 is subtracted from the division of the Force into the Weight. This is to allow for the pull of gravity (1g).

#### Analyzing a Flight

In our lesson Flight Path of a Model Rocket, we measured the maximum velocity of the flight using an electronic altimeter. Using some of the data from that flight we may now determine what the maximum velocity that the rocket should have been using our formula above.

We know that we used a B6-4 motor for the flight and we have in figure 1 above the thrust curve for this motor. From the thrust curve we know that the average thrust of this motor is 5.2N. From the flight analysis we measured the burn time at 0.8s (this is more accurate than using the thrust curve for the burn time). What is left now is to determine the average weight of the rocket.

We measured the mass of the rocket before lift-off at 106g. Using motor specifications from the ThrustCurve.org website we find that the propellent mass of a B6 motor is 6g giving the rocket a mass of 100g after the burn. Taking an average of the two values gives us:

*Mass _{avg}=(106g + 100g) / 2 = 103g*

To determine the average weight of the rocket in newtons we must first convert the mass from grams to kilograms and multiple the result by the gravitational constant of 9.8m/s^{2}:

*Weight _{avg}=(103/1000) * 9.8 = 1.0094N*

Below is a table of all the values we need to determine the maximum velocity:

[ahm-wp-tabular id=4482 template=Web2]

We may now determine the what the theoretical maximum velocity of the flight:

#### Aerodynamic Drag

In the Flight Path of a Model Rocket lesson, we found that the maximum velocity measured was 23 m/s. As we can see this rate is slower than the calculated maximum velocity. Thus we can conclude that aerodynamic drag affected the velocity of the rocket by a difference of 8.8m/s. In our lesson Understanding Rocket Aerodynamics, we discussed the factors that create aerodynamic drag. If we were to apply some of these concepts we would certainly close the gap between the calculated maximum velocity and the actual recorded one.

#### Mathematics in Motion

As we can see rocketry may be used to demonstrate mathematic concepts. It may also be used to demonstrate the effects of aerodynamic drag on moving objects. A good experiment to try would be to use motors with differing average thrust values and compare the results. As well, sanding the fins to be more airfoil shape would be a good experiment to try.

#### Reference Documentation

The formula for velocity used in this article came from the document ‘2844_Estes_Math_of_Model_Rocketry_TN-5’ by Robert L Cannon.

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