Rocket project
Unit Question: How do you describe the maximum height, time, air resistance, and average velocity of flying objects?
This unit's purpose was to teach students to be able to use physics to describe a rocket in flight. To accomplish this we learned about Newton’s laws of motion, types of motion such as linear and 2-dimensional, as well as defining, applying and understanding these concepts through labs, mini projects and lectures over the course of a few weeks. This would allow us to effectively look at certain components of a rocket as well as analyze it while it's in motion on a scientific level of understanding. We also used the engineering design process (EDP) which includes you to ask, research, imagine, plan, create, test, and improve. These processes will encourage students to experience open-ended problem solving of how to create an impressive bottle rocket, learn from failure through tests and refinements, and develop the ability to create innovative solutions to challenges by imagining and planning. furthermore, we used plenty of mathematics to help solve certain parts of our rocket by learning about linear motion and the quadratic formula and how we apply these concepts to rocketry.
Mathematics is also a very crucial part to launching these rockets, specifically quadratics and linear motion. Quadratics can be expressed in the formula of ax²+bx+c where a, b, and c are variables we know of, and x is what we don't know but we can solve for. We use this equation after analyzing the rocket's flight path and collecting data from it and replacing the variables with the new info. We use this equation because on a graph it conveniently is expressed in the same terms as a parabola which is very similar to the projectile's flight path.
The reason that the path of the object creates a parabola is because the velocity of the object can vary, increasing and decreasing the entire time. The formula displays the flying object’s motion in terms of height from the ground at a given time based on the acceleration and gravity. When the projectile is in free fall, we know one force is acting on it; gravity which is about 9.81 meters per second squared. The object can still experience acceleration while in free fall. We have a frame of reference which derives from the data of 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 and its velocity at that point.
Linear motion can be expressed as the motion or displacement of an object in one direction. Linear motion can happen with a constant velocity or varying velocity while in motion. Velocity determines both speed and direction of the displacement of an object which can be written out by the change in position over time of travel. Acceleration represents the change of speed on the graph, acceleration is the graph’s curve. Gs (gravity force) is used to measure acceleration. One G is the equivalent to the acceleration we experience on earth which is about 9.81 meters per second², often calculating with g’s we round it up to 10 meters per second²
Calculations:
After launching our final Rockets it was time to calculate and analyze our results. To assist us in this part during the launch we used an inclinometer that was set 60 M away from the launching pad, multiple video angles, a degree angle measurement to roughly calculate the max height of the rocket, as well as an altimeter to more accurately find the max height during launch.
Max Height:
To determine the max height of our rocket we use trigonometry. Trigonometry is mainly used for obtaining unknown angles and distances from known angles in geometric figures. In our case a triangle is what we're trying to solve. We solve for max height by using SOH CAH TOA (sin, cosine, tangent) but only one of the three functions we use to solve for the height. We already know the measurements of the distance for the launcher —> 60 meters and the height of the inclinometer —> 1.5 meters and 56.13 as our angle. Placing these variables onto an image or a right triangle would look like this:
- What skills did we learn?
- The engineering design process is a cycle and testing and failing until a design works.
- The acceleration due to gravity has a velocity that is constantly changing and can be graphed with a quadratic function.
- Acceleration, Velocity, and position are all related to each other’s slopes on a graph.
- What mini projects did we do in the unit?
- Projectile Motion
- Velocity and Acceleration
- Vectors
- Quadratic Functions
- Factoring
- Completing the square
- Quadratic Formula
This unit's purpose was to teach students to be able to use physics to describe a rocket in flight. To accomplish this we learned about Newton’s laws of motion, types of motion such as linear and 2-dimensional, as well as defining, applying and understanding these concepts through labs, mini projects and lectures over the course of a few weeks. This would allow us to effectively look at certain components of a rocket as well as analyze it while it's in motion on a scientific level of understanding. We also used the engineering design process (EDP) which includes you to ask, research, imagine, plan, create, test, and improve. These processes will encourage students to experience open-ended problem solving of how to create an impressive bottle rocket, learn from failure through tests and refinements, and develop the ability to create innovative solutions to challenges by imagining and planning. furthermore, we used plenty of mathematics to help solve certain parts of our rocket by learning about linear motion and the quadratic formula and how we apply these concepts to rocketry.
Mathematics is also a very crucial part to launching these rockets, specifically quadratics and linear motion. Quadratics can be expressed in the formula of ax²+bx+c where a, b, and c are variables we know of, and x is what we don't know but we can solve for. We use this equation after analyzing the rocket's flight path and collecting data from it and replacing the variables with the new info. We use this equation because on a graph it conveniently is expressed in the same terms as a parabola which is very similar to the projectile's flight path.
The reason that the path of the object creates a parabola is because the velocity of the object can vary, increasing and decreasing the entire time. The formula displays the flying object’s motion in terms of height from the ground at a given time based on the acceleration and gravity. When the projectile is in free fall, we know one force is acting on it; gravity which is about 9.81 meters per second squared. The object can still experience acceleration while in free fall. We have a frame of reference which derives from the data of 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 and its velocity at that point.
Linear motion can be expressed as the motion or displacement of an object in one direction. Linear motion can happen with a constant velocity or varying velocity while in motion. Velocity determines both speed and direction of the displacement of an object which can be written out by the change in position over time of travel. Acceleration represents the change of speed on the graph, acceleration is the graph’s curve. Gs (gravity force) is used to measure acceleration. One G is the equivalent to the acceleration we experience on earth which is about 9.81 meters per second², often calculating with g’s we round it up to 10 meters per second²
Calculations:
After launching our final Rockets it was time to calculate and analyze our results. To assist us in this part during the launch we used an inclinometer that was set 60 M away from the launching pad, multiple video angles, a degree angle measurement to roughly calculate the max height of the rocket, as well as an altimeter to more accurately find the max height during launch.
Max Height:
To determine the max height of our rocket we use trigonometry. Trigonometry is mainly used for obtaining unknown angles and distances from known angles in geometric figures. In our case a triangle is what we're trying to solve. We solve for max height by using SOH CAH TOA (sin, cosine, tangent) but only one of the three functions we use to solve for the height. We already know the measurements of the distance for the launcher —> 60 meters and the height of the inclinometer —> 1.5 meters and 56.13 as our angle. Placing these variables onto an image or a right triangle would look like this:
The frame of reference for this case is 56.13°, so we know the adjacent side to this right triangle and we are trying to solve for the opposite side knowing this we use the trigonometric function tan(θ) = opp/adj replacing the words with our numbers: 60tan(56.13)+1.5=90.98 meters. 90.98 meters would be the theoretical apogee of our rocket’s flight.
Initial Velocity:
To begin finding initial velocity you look at the launcher video of your rocket and count how many frames it took for the rocket to pass the reference post. The reference post was 1.7 m and its purpose was just to find the initial velocity. I counted roughly 3.5 frames. Next was to find how many seconds those frames were. The video was recorded at 30 frames per second so we divided 3.5 frames by 30fps to get 0.1166 seconds. Now you calculate the meter per seconds by taking the reference height divided by the time and call it Vo (starting velocity). Vo = Meters/second —> 1.7/0.1166 = 14.5797 Meters per second is our initial velocity.
Initial Velocity:
To begin finding initial velocity you look at the launcher video of your rocket and count how many frames it took for the rocket to pass the reference post. The reference post was 1.7 m and its purpose was just to find the initial velocity. I counted roughly 3.5 frames. Next was to find how many seconds those frames were. The video was recorded at 30 frames per second so we divided 3.5 frames by 30fps to get 0.1166 seconds. Now you calculate the meter per seconds by taking the reference height divided by the time and call it Vo (starting velocity). Vo = Meters/second —> 1.7/0.1166 = 14.5797 Meters per second is our initial velocity.
Force of Gravity:
The force of gravity is the resultant of two or more forces experiencing an attraction to one another. This is something solved for the rocket’s force of gravity with and without water in our rocket which impacts the mass of our rocket and we use mass in the equation to solve force of gravity (Fg).
With water: Starting by recording the mass of our rocket with water (30% of the pressure chamber filled with water) which was 1.167 kilograms. To find the force of gravity we multiplied the mass (1.161) by the Gravity (9.81), so 1.161*9.81 is 11.4482 N. This means the rocket's force of gravity with water is 11.4482 Newtons.
Without water: The mass of the rocket without water was .242 kilograms. Using the same equations we used to solve for force of gravity with water to get .242 kilograms*9.81 Newtons = 2.374 Newtons (Fg of rocket w/o water). It is possible to use a different equation to find Fg. I used the F = m*a but you could also use the equation A = F/M (Acceleration = Force divided by Mass).
Thrust Force:
Using Vo and the time of takeoff from the video footage we calculated the acceleration during takeoff, we need to find this before finding the thrust because we use acceleration in the next equation, that is used to find the net force, then we can find the thrust.
Acceleration: The equation to find acceleration is a = Vo/t. Inputting our numbers of 14.5797 (Vo) and 0.1166 (t) which is the time it took for the water to expel from the rocket, gives us the answer of 125.0403 m/s². Our acceleration is 125.0403 meters per second².
Net Force: Now that we have acceleration we can find the net force acting on the rocket by multiplying the acceleration the rocket undergoes during takeoff and the mass of it. (I used the mass of the rocket when water isn't in it, our net force would be higher if we solved it with the rocket's mass when it does have water). F net = ma —> .242*125.04003 = 30.25968726 is the net force the rocket has.
Force of thrust: now we can finally find the thrust force of our rocket. To find the force of thrust we add the net force and force of gravity together, We add the net force and force of gravity together to calculate the force of thrust acting on the rocket because gravity is always present and we need to keep it in mind even during the mathematics. The equation looks like this: Ft = F net + Fg —> 30.25968726 + .242 = 30.50168726. The Force of thrust without water is 30.50168726 Newtons.
Theoretical Flight Time:
To find the theoretical flight time we use the quadratic formula. Again, we use the quadratic formula because when we graph it, it follows the same course of a rocket’s flight. The standard form of the equation is ax²+bx+c, we input our numbers into that formula but its formatted different so we know what variables to use; h(t) = -½(g)(t²) + Vo(t) + Yo —> Vo, h(t) ( vertical position giving time), t (time), g = gravity (9.81m/s²) replacing this with numbers we've solved for and were given now looks like this: h(t) = -½(9.81)(t²) + 14.5797(t) + 0.3 = starting height (1.5 ft). The numbers highlighted are our a,b,and c in this equation: x=-b±√b^2-4ac/2a this formula gives t = 2.97. But 2.97, our theoretical time of max height. The theoretical flight time of our rocket would be half our flight time, so roughly 1.472 seconds was the flight time, this isn't accurate to our true flight time because the mathematics did not consider our slow controlled descent, weather conditions, and other variables to our real flight time, it only calculated the rocket going and and straight down with a fast velocity. .
Decent Velocity:
We used our rocket video to calculate the time of touchdown which we will use to find velocity. We did this by counting the frames of the video. It's also possible to find time of touch on by subtracting the calculated time of max height from the frames we counted. We divided the distance traveled (max height) by the time taken calculated in the previous steps. V = d ( 90.83) / t (15) = 5.34 m/s.
The force of gravity is the resultant of two or more forces experiencing an attraction to one another. This is something solved for the rocket’s force of gravity with and without water in our rocket which impacts the mass of our rocket and we use mass in the equation to solve force of gravity (Fg).
With water: Starting by recording the mass of our rocket with water (30% of the pressure chamber filled with water) which was 1.167 kilograms. To find the force of gravity we multiplied the mass (1.161) by the Gravity (9.81), so 1.161*9.81 is 11.4482 N. This means the rocket's force of gravity with water is 11.4482 Newtons.
Without water: The mass of the rocket without water was .242 kilograms. Using the same equations we used to solve for force of gravity with water to get .242 kilograms*9.81 Newtons = 2.374 Newtons (Fg of rocket w/o water). It is possible to use a different equation to find Fg. I used the F = m*a but you could also use the equation A = F/M (Acceleration = Force divided by Mass).
Thrust Force:
Using Vo and the time of takeoff from the video footage we calculated the acceleration during takeoff, we need to find this before finding the thrust because we use acceleration in the next equation, that is used to find the net force, then we can find the thrust.
Acceleration: The equation to find acceleration is a = Vo/t. Inputting our numbers of 14.5797 (Vo) and 0.1166 (t) which is the time it took for the water to expel from the rocket, gives us the answer of 125.0403 m/s². Our acceleration is 125.0403 meters per second².
Net Force: Now that we have acceleration we can find the net force acting on the rocket by multiplying the acceleration the rocket undergoes during takeoff and the mass of it. (I used the mass of the rocket when water isn't in it, our net force would be higher if we solved it with the rocket's mass when it does have water). F net = ma —> .242*125.04003 = 30.25968726 is the net force the rocket has.
Force of thrust: now we can finally find the thrust force of our rocket. To find the force of thrust we add the net force and force of gravity together, We add the net force and force of gravity together to calculate the force of thrust acting on the rocket because gravity is always present and we need to keep it in mind even during the mathematics. The equation looks like this: Ft = F net + Fg —> 30.25968726 + .242 = 30.50168726. The Force of thrust without water is 30.50168726 Newtons.
Theoretical Flight Time:
To find the theoretical flight time we use the quadratic formula. Again, we use the quadratic formula because when we graph it, it follows the same course of a rocket’s flight. The standard form of the equation is ax²+bx+c, we input our numbers into that formula but its formatted different so we know what variables to use; h(t) = -½(g)(t²) + Vo(t) + Yo —> Vo, h(t) ( vertical position giving time), t (time), g = gravity (9.81m/s²) replacing this with numbers we've solved for and were given now looks like this: h(t) = -½(9.81)(t²) + 14.5797(t) + 0.3 = starting height (1.5 ft). The numbers highlighted are our a,b,and c in this equation: x=-b±√b^2-4ac/2a this formula gives t = 2.97. But 2.97, our theoretical time of max height. The theoretical flight time of our rocket would be half our flight time, so roughly 1.472 seconds was the flight time, this isn't accurate to our true flight time because the mathematics did not consider our slow controlled descent, weather conditions, and other variables to our real flight time, it only calculated the rocket going and and straight down with a fast velocity. .
Decent Velocity:
We used our rocket video to calculate the time of touchdown which we will use to find velocity. We did this by counting the frames of the video. It's also possible to find time of touch on by subtracting the calculated time of max height from the frames we counted. We divided the distance traveled (max height) by the time taken calculated in the previous steps. V = d ( 90.83) / t (15) = 5.34 m/s.
Reflection
Overall I think our rocket did quite well. Throughout the entire journey of this course utilizing the engineering design process, launching and analyzing was a success. The prototype and our final design of our rockets remained pretty similar to each other, but also was improved after the opportunity to do so. The final build of our rocket functioned just as we had expected for the launch, rising flight, controlled descent, and landing, especially our controlled descent for a backslider; it was an admirable fall. Taking a new and different route than what most had done over the years with this rocket project, was attempting to use smart water bottles as our final pressure chamber for the rocket, rather than using 2 Liter bottles. This was tedious when it came to splicing because smart water bottle plastic is much thicker than 2 L water bottles as we found. During lunch we also experienced issues with the altimeter causing us to have to launch three times, eventually we figured out our issue was that it wasn't getting enough air to be able read the air pressure as it rose to the sky. Passing on insightful tips to sophomores next year would be to plan and prepare for failure. Although it may not always happen, having a plan B is extremely useful because of the time crunch that we have. Sometimes it's best to use what is left and work with it. Also to be proactive with each portion of the workload but also do a good job to make beautiful work. Procrastination is not going to be useful for any bit of this unit although it may be tempting! If I were to do the project again I’d like to attempt to make a Tommy Toy rocket and rather than using smart water bottles.