TRAIN Scholarship Acceleration Blog Week 10

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Genaro Rivera

November 17, 2022

Accelerometer

Here, I created an equation that represents the forces in the x direction and the y direction for the acceleration of a weather balloon. My team entrusted me with handling the accelerometer. I created a formula to represent the forces in the y-direction. To calculate the upward buoyancy force, you need the formula Fu = (Pf)(V)(g) where Fu is the upward buoyancy force, Pf is the air density, V is the volume, and g is the acceleration due to gravity which on earth is always 9.8 m/s^2. I did the calculations and got the ay equation you see above. I kept Pf and R as variables because because the air density changes as you go up, and the balloon increases in volume as you go up as well. Additionally, I defined the downward force as Fd = ma. which is newton’s second law of motion and 6.3896 newtons of force downwards. I created the free body diagram. In the free body diagram, I included the force of the wind. The weather news said that the wind will be 4 mph or 1.79 m/s. I used 1.79 m/s. I created a right triangle to represent the force in the x direction and the force in the y direction. Using SOH CAH TOA, I solved for x and y. I solved for acceleration in the y direction. To solve for acceleration in the y direction I simply added up all the forces in the y direction which included the upward force, downward force, and the wind force in the y direction. Afterwards, I replaced F with ma since newton’s second law states that F=ma. I then dived both sides by m. The mass of the payload is 0.6520 kg. I solved for acceleration in the x direction. The only force in the x direction would be the wind. I set Fx = 1.79cos(theta). Afterwards, I replaced Fx with ma. I solved an got ax equation above. The sign can vary based on what direction the wind is blowing in, but the magnitude is the same in all directions.

To ensure my equations made sense, I integrated them twice to get

I then plugged-in values for the speed in the Y-direction assuming the balloon was halfway through the flight.

Upon doing so, the distance was 233,910 feet. The highest a weather balloon has gone is 117,127 feet Arizona Near Space Research 2). My formula included the trip back down so that would be 200,000-300,000 feet, so it’s reasonable to say my formulas are good.

My main concern with the acceleration formula is to see the relationships between each variable and the acceleration to make predictions.

Side Note: Pf = P/(RT) where P is pressure, T is temperature, and R is a constant (the universal gas constant)

Relationships:

•       The air density, radius of balloon, and the angle at which the wind blows are directly proportional to the acceleration.
Side Note to show consistency: Mass (0.6520 kg) is inversely proportional to acceleration.

•       Pressure is directly proportional to acceleration while temperature is inversely proportional to acceleration.

•       Acceleration in the x direction is only dependent on the angle at which the wind pushes it.

Conclusion:

•       The radius will increase at a constant rate causing the acceleration to increase which can be modeled by the equation y = (1/8000)x + 12.5 where x is the height in feet and y is the radius in feet.

•       However, according to a team member’s predictions, the pressure will decrease as you go up. The pressure decreases in an exponential way, so it will outweigh the increasing rate of the balloon. Eventually, the pressure will reach close to zero, and according to the equation, the acceleration will reach close to 0.

•       According to another team member’s predictions, the temperature will decrease and then increase.

•       So, my predictions are that the acceleration in the y direction will increase at an increasing rate before increasing at a decreasing rate. Around when the temperature starts to increase, the acceleration will start to decrease at an almost constant rate before popping and then traveling at a constant acceleration until it hits the ground.

 

 

Sources

Arizona near space research (no date) Arizona Near Space Research - About. Available at: https://ansr.org/About-Us (Accessed: November 17, 2022).

 

 




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