Adventure 10 — Activity

🧩 Part 1 — Mission rule

Look back at the distance curves above

The mission requires Feynman to stay at least twice as far from Earth as Gamow.

1A) Write the mission rule using a ratio

1B) What does this rule mean in words?

Use the chart and formulas to identify the velocity and acceleration functions.

2A) What is f'(t)?

2B) What is g'(t)?

2C) What is f''(t)?

2D) What is g''(t)?

Use the distance chart above

Fill in the table from the chart or by calculation. Then check your answers.

Use f(t)=2t²+175t f'(t)=4t and g(t)=t²+5t g'(t)=2 to Fill in the distance and velocity ratio tables.

What do you notice?

4A) What is f''(t)/g''(t)?

4B) As t → ∞, what value do the distance and velocity ratios approach?

Use one or two sentences to explain why the distance ratio approaches 2.

5A) Explanation

5B) Challenge: for same-degree polynomials, what does the ratio approach?

The mission rule is a ratio statement: Feynman must stay at least twice as far from Earth as Gamow.
The second derivatives are constants, so their ratio is the easiest place to start.
The first table values are 925/50, 1950/150, 4300/500, 7050/1050, and 13750/2750.
The long-time table shows the same pattern in both distance and velocity ratios: both approach 2.
At long times, same-degree polynomials are controlled by their leading terms.

Sample answers (for checking)

f(t)/g(t) ≥ 2 for all t ≥ 0.
f'(t)=4t+175, g'(t)=2t+5, f''(t)=4, g''(t)=2.
Table 1: (925, 50, 18.50), (1950, 150, 13.00), (4300, 500, 8.60), (7050, 1050, 6.71), (13750, 2750, 5.00).
Table 2: (2.67, 2.34), (2.23, 2.11), (2.14, 2.07).
f''(t)/g''(t)=2, and both long-time ratios approach 2.
For same-degree polynomials, the ratio approaches the ratio of the leading coefficients.