Physics of Apollo 11 and Shredding Costs

Apollo 11 Crew G-Force

• The Apollo 11 crew experienced a force of 1.2g at liftoff.
• The maximum force experienced was 3.9g.
• The g-force is measured in Newtons (N).
• 1N is equivalent to kg(m/s²).
• p(500) = 3.2 means at a height of 500 meters, the astronauts experienced a force of 3.2 Newtons.
• p'(500) = 0.2 means at a height of 500 meters, the force on the crew is increasing at a rate of 0.2 Newtons per meter.
• p'(h) = -1 means as the height (h) increases, the rate of the force on the crew decreases at a rate of 1 Newton per meter.

Law Firm Shredding Costs

• The cost of shredding documents (C) is a function of the weight of paper (w) in pounds.
• C = f(w) represents the cost function.
• f(65) = 45 means the cost to shred 65 pounds of paper is $45.
• f'(80) = 1.25 means after shredding 80 pounds of paper, the cost per pound of shredding is increasing at a rate of $1.25 per pound.
• f'(200) = 2.5 means when you shred 200 pounds of paper, the cost of shredding 1 additional pound of paper before the cost of shredding decreases by $2.50.

Antibiotic Dosage

• The dosage of an antibiotic (D) is a function of the patient's weight (w) in pounds.
• D = f(w) represents the dosage function.
• f(135) = 120 means a patient weighing 135 pounds needs 120 milligrams of the antibiotic.
• f'(135) = 2.5 means for a patient weighing 135 pounds, their antibiotic dosage increases by 2.5 milligrams for every 1 pound increase in weight.

The Derivative in Context

• The derivative can be interpreted as the instantaneous rate of change at a specific point.
• The Leibniz notation, 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑, helps visualize the derivative as the limit of ratios, representing a small change in a process.
• The derivative can be expressed as 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑, where "𝑑𝑑𝑑" represents a small difference in the process and 𝑑𝑑𝑑𝑑 represents the change in the independent variable.

Interpreting Derivatives in Real-World Scenarios

• Example 1: The derivative 𝑑𝑑𝑉𝑉/𝑑𝑑𝑡𝑡=−2 at 𝑡𝑡=3 means that the volume of the fish tank is decreasing at a rate of 2 gallons per minute at the 3rd minute.

Using Proper Notation with Variables

• Example 2:
  • Scenario A: The cost 𝐶𝐶 to extract a barrel of oil is decreasing at a rate of $3 per barrel, expressed as 𝑑𝑑𝐶𝐶/𝑑𝑑𝑏𝑏 = −3, where 𝑏𝑏 represents the number of barrels extracted.
  • Scenario B: The rate of oil leakage from the Deepwater Horizon oil spill at the end of the first week was 790 cubic meters per day. This information can be written as 𝑑𝑑𝑉𝑉/𝑑𝑑𝑡𝑡 = 790, where 𝑉𝑉 represents the volume of oil spilled and 𝑡𝑡 represents time in days.
  • Scenario C: The population of mountain gorillas increased by 114 gorillas in 2003. This can be written as 𝑑𝑑𝑃𝑃/𝑑𝑑𝑡𝑡 = 114, where 𝑃𝑃 represents the gorilla population and 𝑡𝑡 represents time in years.

Interpreting Derivatives in Alien Scenarios

• Example 3:
  • Scenario A: 𝑃𝑃′ 3 = 0.5 means that the alien population on Newtonia is increasing at a rate of 0.5 million aliens per year, 3 years after the planet’s discovery.
  • Scenario B: 𝑃𝑃−1 7.5 = 5 means that 7.5 years after Newtonia’s discovery, the alien population was 5 million.
  • Scenario C: 𝑃𝑃−1 7.5 ′ = 0.75 means that the alien population was increasing at a rate of 0.75 million aliens per year 7.5 years after the planet’s discovery.

Using Units to Interpret Derivatives

• Example 4: 𝑑𝑑𝑠𝑠/𝑑𝑑𝑡𝑡 represents the velocity of the water balloon, which is the rate of change of distance with respect to time. The units of 𝑑𝑑𝑠𝑠/𝑑𝑑𝑡𝑡 would be feet per second.

Explore the physics concepts behind the Apollo 11 crew's experiences with g-forces during liftoff and ascent. Additionally, delve into the cost functions related to shredding documents based on weight. Test your understanding of these principles in this engaging quiz.