# Technical Note T -5 Elementary Mathematics of Model Rocket Flight

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1 Technical Note T -5 Elementary Mathematics of Model Rocket Flight By Robert L. Cannon Updated and edited by Ann Grimm EstesEducator.com Estes-Cox Corp.

2 This publication contains three separate articles on different aspects of model rocket flight. One article explains how the heights reached by model rockets can be determined. Instructions on building and using your own altitude determining device are included, as well as an elementary explanation of the theory involved. The second article provides facts and examples for helping the students build better concepts about relative velocities and speeds. The final article about acceleration provides more advanced information on speed, acceleration and distance traveled per unit of time. EST #844 Copyright 1970, Centuri Corporation. All rights reserved.

7 Acceleration is the process of speeding up. If something begins to go faster it accelerates. Your model rocket sitting on the launch pad is not accelerating. When it starts to move, it accelerates. As long as its speed is increasing, the rocket is accelerating. To be technical, an object which is gaining speed is undergoing positive acceleration. A moving body which is slowing down is undergoing negative acceleration. Negative acceleration is sometimes called deceleration. No Acceleration Positive Acceleration Negative Acceleration A good example of acceleration occurs when you throw a ball straight up into the air. The ball is going very fast as it leaves your hand. What are the two forces acting on the ball to slow it down? Two forces slowing down the ball: 1.. The two forces are gravity and drag (friction between the ball and the air through which it traveling). The ball soon loses all of its momentum, stops going up and starts to fall back to the ground. The rate at which the ball gains speed (acceleration) as it falls is about 3 feet per second per second (3 ft/sec ). This expression (3 feet per second per second) means that a falling object near the ground falls 3 feet per second faster for each second that it falls. In other words, a ball dropped from the top of a tall building will fall 16 feet during its first second of fall. Does this surprise you? Let s take an example to help us understand this. If a car travels for one hour and begins its trip from a standing start and very slowly and steadily accelerates until it is traveling at a rate of 60 miles per hour, what was the car s average speed during this hour of travel? Average Speed = final speed + original speed How far does a falling object travel during two seconds? Average Speed Distance Traveled = final speed + original speed = 64 feet per second + 0 = 3 feet per second = Average speed x Time in motion = 3 feet per second x seconds = 64 feet How far did the falling ball travel during the second one-second of its fall? Distance Traveled in the Second Second = Total distance traveled - Distance traveled in first second To simplify calculations we can use formulas instead of having to think through all of the steps in a problem each time we need to solve another problem of the same type. For example, the formula for determining the average speed of a body is-- v = v + v 1 v v = 64 feet - 16 feet = 48 feet = average velocity = final velocity (speed) Average Speed = v 1 = original velocity The car s average speed was miles per hour or 30 miles per hour. The formula for determining the distance an object falls during a given time is-- s = 1/ g t Once the ball leaves your hand it does not go any faster, in fact, the ball start slowing down as it rises. A car traveling at the average rate of 30 miles per hour for one hour travels 30 miles in that hour. A ball accelerating at the rate of 3 feet per second travels 16 feet in the first second of its fall. s g t = distance = acceleration due to gravity = time 7

8 How far does a falling body travel during the third second of its fall? s = 1/ gt s = 1/ x 3 feet/second x (3 seconds) = 1/ x 3 feet/second x 9 seconds = 1/ x 3 feet x 9 = 16 x 9 feet = 144 feet The ball falls 144 feet in three seconds. Since the ball fell 64 feet in two seconds, the ball falls 80 feet during the third second of its fall (144 feet minus 64 feet). A falling object keeps accelerating because of the force of gravity acting on it until the friction of the air moving past the falling body prevents the object from falling any faster. When this maximum speed is reached the object ceases to accelerate and falls at its terminal velocity. LET S TRY A PROBLEM Using the following formulas (some are simplified to avoid using higher mathematics) we can determine some values for accelerations and velocities for model rockets. The values given are based on theoretical no drag conditions. v = F ( 1) w av ALPHA FLIGHT ANALYSIS Using the above formula for velocity, determine the burnout velocity of an Alpha launched using an A8-3 engine. The Alpha weighs 0.8 ounces without an engine. With an A8-3 engine the Alpha weighs 1.37 ounces at liftoff. The weight of propellant in an A8-3 engine is 0.11 ounces giving an average weight of 1.3 ounces during the thrust phase of the flight. The A8-3 engine thrusts for 0.3 seconds and has a total impulse of 0.56 pound-seconds. (These values may be found in or calculated from information in the current Estes catalog.) First find the average force, then use this force in the velocity formula to find the final velocity. Force = (thrust) gt w av = average weight of rocket F = force (average thrust of rocket engine) g = acceleration due to gravity (3 ft./sec. ) t = time in seconds total impulse burn time x 16 oz. 1 lb. } conversion factor Distance inf Feet v = A graph can present a lot of information quickly. Studying a graph can sometimes help you to understand something that may be hard to understand otherwise. Examine the graph for the distances a falling body travels during the first few seconds of its fall. This graph represents the distance a body falls under the constant acceleration of gravity (neglecting air friction) DISTANCE TRAVELED BY A FALLING BODY oz. ( -1) 1.3 oz. Time in Seconds F = 0.56 lb. - sec. 16 oz. x 0.3 sec. 1 lb. = 1.75 x 16 oz. = 8.0 oz. v = F ( 1) w av = (1.1-1)(3ft./sec. )(0.3 sec.) = (0.1) (10.4 ft./sec.) = ft./sec. (3 ft./sec. )(0.3sec.) ( ) F w 1 The expression av gives you an idea of the number of gravities the rocket experiences in upward flight during acceleration. The 1 is subtracted in this expression to allow for the pull of Earth s gravity on the rocket. This rocket develops a fairly high velocity by the end of the thrusting phase of flight. What will be the velocity developed by the same Alpha launched with a C6-5 engine? gt GRAPHS To help understand the velocities which falling objects can develop, (neglecting air friction) examine a graph of the velocities developed by freely falling objects. These graphs are based on results obtained by use of these formulas- Velocity In Feet Per Second v = a t s = v 1 t + at a = acceleration This velocity ( feet per second) developed by the Alpha with the C6-5 engine is more than triple the velocity (06.95 feet per second) which was developed by the A8-3 engine. Notice that the thrust and therefore the acceleration in g s produced by the C6-5 engine (13.17 g) is less than that produced by the A8-3 engine (0.1 g), but the maximum velocity is greater when using the C6-5 engine because the burn time for this engine is greater. I wonder what the acceleration would be in g s for the Big Daddy with a D1-5 engine? Hmm... Reprinted from MRN Vol. 10, No.. VELOCITY OF A FALLING BODY Time in Seconds To find the actual velocities which your birds will attain when allowance for air drag is made, use the Estes book Altitude Prediction Charts (EST 84). F = = = v = = =.5 lb./sec. 16 oz. x 1.70 sec. 1 lb. ( -1) 1.49 oz. = ft./sec. (3ft./sec. ) (1.7sec.) 8

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Making Paper Rockets Description: Students will construct paper rockets and launch them with a commercially available foot-pump rocket launcher or an industrial strength rocket launcher built by the teacher