For this rocket project, students were asked to build functional rockets with safe launch and descents out of everyday items. The goal was to learn how to analyze a rocket’s flight in terms of the laws of physics and the quadratic formula; while also exploring the engineering and design process (EDP) which consists of a series of steps that an engineer takes when creating a final design and product by refining a product. The purpose of the EDP is to give an individual a way to define a problem and then find solutions to the problem through brainstorming, refinement, testing, and improvement. In order to prepare for our final project, we spent multiple weeks learning about linear motion and the quadratic formula and how they relate to rocketry, while also becoming experts on Newton’s laws of motion and different scientific equations to find things like force, mass, acceleration, and velocity.
The basic quadratic equation is written in the form of ax²+bx+c where a, b, and c represent known numbers, and x represents the unknown. When analyzing the flight path of a rocket, we can plug the data into this formula. In terms of quadratic expressions, the reason that we can use them to find the height and time of a rocket’s flight is because graphically, a quadratic equation follows a parabola, which is similar to a rocket’s flight or any projectile. The formula represents a predictable path of projectile motion in terms of height from the ground at a given time due to gravity and acceleration. Acceleration is the change in velocity of an object. This is represented by the changing slope throughout the graph. During freefall, an object is only being acted upon by the force of gravity which is about 9.8 meters per second squared and nothing else. The object can still experience acceleration at this time. The reason that the path of a projectile creates a parabola is because the velocity of the object is constantly changing. We use a frame of reference off of which we base our data collection by translating data from our graph into the real-world scenario. By analyzing things such as points on the graph, we can determine the position of our rocket during its flight. The slope at each point represents the rocket’s velocity.
Linear motion describes the one-dimensional path of an object’s displacement or motion. Linear motion can be in the form of an object moving with constant velocity or changing velocity. Velocity describes the direction of movement and speed of an object. Acceleration describes the rate of change in velocity of the object. Gs are used to measure acceleration. One G equals the acceleration experienced due to gravity, which on Earth is about 9.806 meters per second squared (m/s^2) which we can round up to 10 m/s^2.
Calculations:
Throughout the project, we practiced doing calculations to assist us when analyzing the flight of our rocket. We measure the apogee, or max height of our rocket’s flight by using an inclinometer that was set up 60 meters away from the launch pad and multiple video angles. We also measured the altitude with more accuracy with an altimeter that was strapped into a chamber in the rocket.
Max Height: We utilized trigonometry to calculate the max height. Because we already knew the angle and adjacent side, we could use this data to calculate the missing side length. These known variables were 56.13° as our angle and 60 meters as our adjacent side to the right angle. I plugged these numbers in using a tangent → tan(θ) = opp/adj in the form of 60tan(56.13)+1.5=90.98 meters. 90.98 meters would be the theoretical apogee of our rocket’s flight at 1.472 seconds .
Initial Velocity: The first step for finding initial velocity was to watch the launch and count the number of frames of the video it took for the rocket to pass the reference post. The video runs at 60 frames per second, so to convert the frames into seconds we needed to divide the number of frames divided by 30 which gave us 0.1166. 1.7m was the height of the reference post, so we then divided 1.7 by 0.1166, going off of the equation meters/second=initial velocity. This means that the initial velocity would be 14.5797 meters per second (m/s).
Force of Gravity: Objects near earth experience acceleration at the rate of 9.81 m/s/s. We can round that up to 10. We recorded the mass of our rocket with water (1.167) and without water (0.242). We chose to focus on the mass with water. The force-mass equation is force equals mass times acceleration (f=ma). We can replace acceleration with 9.81 or ten m/s/s as gravity. Plugging in 1.167 times 9.81 m/s/s which equals 11.448 (full) or 2.374 (empty).
Thrust: Using our initial velocity (14.5797 m/s) divided by the time of takeoff (1.166), we calculated the acceleration of the rocket during flight which was about 125.0403. Then, we calculated the net force acting on the rocket by plugging our data into the force-mass equation which is mass times acceleration. 125.0403 times 0.1166 = 145.922. Finally, we calculated the force of thrust acting on the rocket by adding the force of gravity to the net force: 145.922+9.81=155.732.
Theoretical Flight Time: The quadratic equation that we used to find this was in the form of x=-b±√b^2-4ac/2a. Our height given time formula was ½(9.81)(t^2)+14.5797(t)+0.3. We plugged our height given time formula into the quadratic formula using our quadratic formula calculator. Zero #2 represents the theoretical total flight time which would be 2.964. About ½ of this number gives us the theoretical time of max height which is 1.472 seconds. This is not really accurate, as it is only telling us the time if it had gone straight up and back down at the same velocity in the best conditions. We can picture this in the parabola of the rocket’s flight on a graph. On the graph, our rocket's flight looks like a perfect symmetrical arch. However an accurate representation of this on the graph would actually look something like this:
This is because the curve represents a constantly changing velocity. However, as the rocket descends in a controlled manner it begins to fall at a constant rate.
Descent Velocity: To calculate our descent velocity, we used our rocket video to find the time of touchdown. We did this by subtracting the time of max height from the time of touchdown. The descent time was 17 seconds, and the time of max height was 5 seconds. We then divided the max height by the amount of time taken. This gave us 5.34 m/s.