rocket project math portion
COVER LETTER
The goal of this project was to make a successful bottle rocket that had to include at least 1 fin, an altimeter chamber, a parachute deployment system, a pressure chamber, and a some sort of nose cone. We also got to experience the trial and error processes, when using the steps of the engineering design process. The engineering design process is the engineering design cycle. The stages include Ask, Research, Imagine, Plan, Create, Test, and Improve. I broke it into 3 parts, the first sages including Ask, Research, and Imagine this when you are asking questions finding out what would work best to maximus the rocket height. Next is the Plan and Create stage where you have your “final'' plan on paper and your material is ready to go. Lastly is the Test and Improve stage where you are making your final rocket and testing and refined the rocket finding out what worked and what didn't. I believe that all these steps help our group work together smoother and we were really able to see each other strengths and weaknesses, as we worked through failure and achievement theses steps were a critical part of the rocket launch processes.
When a rocket is launch there is a mathematical equation called a quadratic is aax2+bx+c which is used to find the values of the unknown variable x. We also went over how to get the slope which is crucial for the velocity. We know that velocity is represented by the slope of each point (slope=velocity). But we also needed to understand how position is represented given height vs time. The definition we used was represented by a point where its rocket’s height given time. And lastly was acceleration and how it is represented the change in velocity and acceleration can be seen as the curve itself. A rocket's flight is drawn as a curve and a quadratic is a function where the graphs make that curve. We also learned of to draw free body diagrams which is very helpful way to show the type of forces used in the specific movement, as well as frame of reference which is a set of criteria or values in relation to which measurements can be made from, we used this to help calculate the time our rocket was in the air. Lastly we had to understand free fall which is a downward movement under the force of gravity only, but we had to stop this by making a parachute deployment system
Lastly we had to understand all of the newton's laws of physics first we had to understand his first law Newton’s First Law – Inertia which states “every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force”. Meaning that our rocket would remain on the launch pad until propelled upward, but there is also normal force which is where normal and gravity balance out, finally net force the net force is zero if the forces are in equilibrium. If they are not, then we have a net force, and the object changes motion. Next we have Newton’s Second Law – Force and Mass force = mass x acceleration, when launching a rocket you see many types of forces presentation the rocket, once the rocket is moving upward you see that there would be drag behind and friction is found between the air surface and the top of the rocket, lastly we must explain the difference between mass and weight, Mass is a measure of inertia that more mass of object has the more inertia and weight is where if the rocket has experiences a much greater force. But because of the bigger mass, it would resist acceleration more. The last law of physics is one of most known being Newton’s Third Law – Action and Reaction: stating that for every action (force) in nature there is an equal and opposite reaction. Meaning that forces always occur in pairs, one body cannot exert a force on another without experiencing a force itself, example is the water pushing on the rocket, the rocket pushing the water shooting it up-wards then falling to the and the ground pushing on the rocket, and rocket pushing on the ground
The goal of this project was to make a successful bottle rocket that had to include at least 1 fin, an altimeter chamber, a parachute deployment system, a pressure chamber, and a some sort of nose cone. We also got to experience the trial and error processes, when using the steps of the engineering design process. The engineering design process is the engineering design cycle. The stages include Ask, Research, Imagine, Plan, Create, Test, and Improve. I broke it into 3 parts, the first sages including Ask, Research, and Imagine this when you are asking questions finding out what would work best to maximus the rocket height. Next is the Plan and Create stage where you have your “final'' plan on paper and your material is ready to go. Lastly is the Test and Improve stage where you are making your final rocket and testing and refined the rocket finding out what worked and what didn't. I believe that all these steps help our group work together smoother and we were really able to see each other strengths and weaknesses, as we worked through failure and achievement theses steps were a critical part of the rocket launch processes.
When a rocket is launch there is a mathematical equation called a quadratic is aax2+bx+c which is used to find the values of the unknown variable x. We also went over how to get the slope which is crucial for the velocity. We know that velocity is represented by the slope of each point (slope=velocity). But we also needed to understand how position is represented given height vs time. The definition we used was represented by a point where its rocket’s height given time. And lastly was acceleration and how it is represented the change in velocity and acceleration can be seen as the curve itself. A rocket's flight is drawn as a curve and a quadratic is a function where the graphs make that curve. We also learned of to draw free body diagrams which is very helpful way to show the type of forces used in the specific movement, as well as frame of reference which is a set of criteria or values in relation to which measurements can be made from, we used this to help calculate the time our rocket was in the air. Lastly we had to understand free fall which is a downward movement under the force of gravity only, but we had to stop this by making a parachute deployment system
Lastly we had to understand all of the newton's laws of physics first we had to understand his first law Newton’s First Law – Inertia which states “every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force”. Meaning that our rocket would remain on the launch pad until propelled upward, but there is also normal force which is where normal and gravity balance out, finally net force the net force is zero if the forces are in equilibrium. If they are not, then we have a net force, and the object changes motion. Next we have Newton’s Second Law – Force and Mass force = mass x acceleration, when launching a rocket you see many types of forces presentation the rocket, once the rocket is moving upward you see that there would be drag behind and friction is found between the air surface and the top of the rocket, lastly we must explain the difference between mass and weight, Mass is a measure of inertia that more mass of object has the more inertia and weight is where if the rocket has experiences a much greater force. But because of the bigger mass, it would resist acceleration more. The last law of physics is one of most known being Newton’s Third Law – Action and Reaction: stating that for every action (force) in nature there is an equal and opposite reaction. Meaning that forces always occur in pairs, one body cannot exert a force on another without experiencing a force itself, example is the water pushing on the rocket, the rocket pushing the water shooting it up-wards then falling to the and the ground pushing on the rocket, and rocket pushing on the ground
Blue print |
Reflection
This project was one of the projects that I think defines AHS, we were hands on learners and also got a better understanding of trial and error and refinement. For this project I worked with a group that i usually don't work with because i wanted to expand who i can be partens the at first i was nervous that I was gonna do all the work, but in the end it really worked out in my favor our rocket was this year winning rocket, our rocket went a total of 341 feet in the air, because of the way we worked together as a group. At first i felt like i was doing all the work and this is how it was how it was gonna be throughout the whole project but I quickly realized when we started building our rocket that I have lots and lots of help from my teammates, and we each had our strength and weaknesses while iI was good at the planning and writing aspects Joseba was really good the building and cutting and max was really good at filling in where ever needed.
We took a lot of risks in this project including splicing our rocket. The risk when spiclicing it is exploding when pumping air into it. We had to ensure our slice was perfect and there were no holes or gaps where air could escape, lucky our rocket didn't explode and the splice was very beneficial. Another risk we took was adding a back slider. We had to make sure it was balanced so we would glide back and forth slowing the descent. We spent hours perfecting the back slider and when we did the final launch both the back slider and splices worked!!
Though I feel like I was the leader in the project I usually do take the leadership role but this time I didn't feel too bossy or too in charge I was just able to direct our group so we got everything done on time and we did end up winning for heightes rocket.
This project was one of the projects that I think defines AHS, we were hands on learners and also got a better understanding of trial and error and refinement. For this project I worked with a group that i usually don't work with because i wanted to expand who i can be partens the at first i was nervous that I was gonna do all the work, but in the end it really worked out in my favor our rocket was this year winning rocket, our rocket went a total of 341 feet in the air, because of the way we worked together as a group. At first i felt like i was doing all the work and this is how it was how it was gonna be throughout the whole project but I quickly realized when we started building our rocket that I have lots and lots of help from my teammates, and we each had our strength and weaknesses while iI was good at the planning and writing aspects Joseba was really good the building and cutting and max was really good at filling in where ever needed.
We took a lot of risks in this project including splicing our rocket. The risk when spiclicing it is exploding when pumping air into it. We had to ensure our slice was perfect and there were no holes or gaps where air could escape, lucky our rocket didn't explode and the splice was very beneficial. Another risk we took was adding a back slider. We had to make sure it was balanced so we would glide back and forth slowing the descent. We spent hours perfecting the back slider and when we did the final launch both the back slider and splices worked!!
Though I feel like I was the leader in the project I usually do take the leadership role but this time I didn't feel too bossy or too in charge I was just able to direct our group so we got everything done on time and we did end up winning for heightes rocket.
Calculations:
Calculate Max Height Using SOH CAH TOA
The goal of the math portion was to figure out the max height of your rocket.
When calculating your rocket total flight time you use a the SOH CAH TOA formula to find the total height and flight time for your rocket in this case we used tan to find the length of the triangle side next to the angle with a given measure, provided that the side across the angle has a known measure as well
60 tan(60.4)=x/60 ⋅ 60 x=60 tan (60.4) x=105.619+1.5
Initial Velocity
To find your initial velocity without a video you need to calculate
h(t)= height given time
g= gravity (9.81)
Vo= velocity
Ho=starting height (0.3)
We used the formula h(t) = -½ (g) (t2) + V0(t) + y0
H(t) = -½ gt Vot +Ho H(t) = -5t^2 Vot +0.3
105 =-5 (4.65)^2 +Vo (4.65)+0.3
105=-108.1125+Vo(4.65)+0.3
-6.3 -6.3
104.7=-108.1125+Vo(4.65
+108.1125 +108.1125
212.8125 Vo(4.65
—----------- = —--------- = 45.763
4.65 4.65
So our velocity would be 45.763
Calculating Theoretical Flight Time using the Quadratic Formula
We used the height given time formula h(t) and the quadratic formula to calculate the theoretical flight time.
Where:
V0 (initial velocity) = 45.763
h(t) (max height over time) = 241
t (time of max height) = 4.65
g (gravity) =
Yo (starting height) = 1.5
Using the quadratic formula
-bb2-4ac2(a)
We are now able to put A^2 +b =c^2 into the formula to find the theoretical flight time
-12(9.81) (+)+45.7(t) + 0.3)
—--A—--- —----b—-- —--c—--
t=-(4.57)+(45.7) 2+(4.9) (0.32(9.81) t=-(4.57)-(45.7) 2+(4.9) (0.32(9.81)
t= 0.00
t=9.146 So our over all total flight time was 9.146sec
Force of Gravity with and without water
All of earth objects accelerate at the same rate 9.81m/s2 because of this we can calculate the force of gravity acting on each object so in this case we need to find the force of gravity with and without water
The mass of our rocket with water
1.742 x gravity (9.81)
The mass without water
0.273 x gravity (9.81)
Next we needed multiply the mass by the force of gravity so mass x 9.81
Fg full =mg =17.089 Fg empty= mg =2.678
Thrust Force
In order to calculate the force of thrust on a rocket we needed to first identify the average rate of acceleration of the rocket during launch. Once we do this we can find the net force acting on the rocket.
Force-mass equation
F=ma
Average acceleration equation
A = Δ v/ t
Time of takeoff divided by acceleration during takeoff
a=Vo/T = 152
Net force x the acceleration during takeoff x mass of the rocket
F net =ma = 41.49
Thrust force acting on rocket = gravity of net force
Ft =F net+Fg =44.168
Descent Velocity
In order to find the average decent velocity of the rocket we must assume that it descend at a constant rate from max height
Average velocity equation
v=d/t
Frames from touchdown= 16 seconds
Max height - time of touch down =12 seconds
v=d/t=8.255
But why does the rocket only follow a parabolic trajectory
Velocity is constantly changing
What shape should it follow after the descent has begun
A straight line
Calculate Max Height Using SOH CAH TOA
The goal of the math portion was to figure out the max height of your rocket.
When calculating your rocket total flight time you use a the SOH CAH TOA formula to find the total height and flight time for your rocket in this case we used tan to find the length of the triangle side next to the angle with a given measure, provided that the side across the angle has a known measure as well
60 tan(60.4)=x/60 ⋅ 60 x=60 tan (60.4) x=105.619+1.5
Initial Velocity
To find your initial velocity without a video you need to calculate
h(t)= height given time
g= gravity (9.81)
Vo= velocity
Ho=starting height (0.3)
We used the formula h(t) = -½ (g) (t2) + V0(t) + y0
H(t) = -½ gt Vot +Ho H(t) = -5t^2 Vot +0.3
105 =-5 (4.65)^2 +Vo (4.65)+0.3
105=-108.1125+Vo(4.65)+0.3
-6.3 -6.3
104.7=-108.1125+Vo(4.65
+108.1125 +108.1125
212.8125 Vo(4.65
—----------- = —--------- = 45.763
4.65 4.65
So our velocity would be 45.763
Calculating Theoretical Flight Time using the Quadratic Formula
We used the height given time formula h(t) and the quadratic formula to calculate the theoretical flight time.
Where:
V0 (initial velocity) = 45.763
h(t) (max height over time) = 241
t (time of max height) = 4.65
g (gravity) =
Yo (starting height) = 1.5
Using the quadratic formula
-bb2-4ac2(a)
We are now able to put A^2 +b =c^2 into the formula to find the theoretical flight time
-12(9.81) (+)+45.7(t) + 0.3)
—--A—--- —----b—-- —--c—--
t=-(4.57)+(45.7) 2+(4.9) (0.32(9.81) t=-(4.57)-(45.7) 2+(4.9) (0.32(9.81)
t= 0.00
t=9.146 So our over all total flight time was 9.146sec
Force of Gravity with and without water
All of earth objects accelerate at the same rate 9.81m/s2 because of this we can calculate the force of gravity acting on each object so in this case we need to find the force of gravity with and without water
The mass of our rocket with water
1.742 x gravity (9.81)
The mass without water
0.273 x gravity (9.81)
Next we needed multiply the mass by the force of gravity so mass x 9.81
Fg full =mg =17.089 Fg empty= mg =2.678
Thrust Force
In order to calculate the force of thrust on a rocket we needed to first identify the average rate of acceleration of the rocket during launch. Once we do this we can find the net force acting on the rocket.
Force-mass equation
F=ma
Average acceleration equation
A = Δ v/ t
Time of takeoff divided by acceleration during takeoff
a=Vo/T = 152
Net force x the acceleration during takeoff x mass of the rocket
F net =ma = 41.49
Thrust force acting on rocket = gravity of net force
Ft =F net+Fg =44.168
Descent Velocity
In order to find the average decent velocity of the rocket we must assume that it descend at a constant rate from max height
Average velocity equation
v=d/t
Frames from touchdown= 16 seconds
Max height - time of touch down =12 seconds
v=d/t=8.255
But why does the rocket only follow a parabolic trajectory
Velocity is constantly changing
What shape should it follow after the descent has begun
A straight line