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By Benjamin Crowell

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6 Conservation of energy tells us 0 = ∆E = ∆K + ∆Eheat . In the plane’s frame of reference, the air’s initial velocity is vi =270 m/s, and its final velocity is zero, so the change in its kinetic energy is nega5 Syst`eme International It’s not at all obvious that the solution would work out in the earth’s frame of reference, although Galilean relativity states that it doesn’t matter which frame we use. Chapter 3 discusses the relationship between conservation of energy and Galilean relativity. 1 Energy 39 tive, ∆K = Kf − Ki = 0 − (1/2)mvi 2 = −(1/2)mvi 2 .

The plane climbs up high, and then drops straight down like a rock, and since the people are falling with the same acceleration as the plane, the sensation is just like what you’d experience if you went out of the earth’s gravitational field. If the plane can start from 10 km up, what is the maximum amount of time for which the dive can last? Based on data about acceleration and distance, we want to find time. Acceleration is the second derivative of distance, so if we integrate the acceleration twice with respect to time, we can find how position relates to time.

Vi = c∆T 2 Note how the mass cancels out. This is a big advantage of solving problems algebraically first, and waiting until the end to plug in numbers. With a purely numerical approach, we wouldn’t even have known what value of m to pick, or if we’d guessed a value like 1 kg, we wouldn’t have known whether our answer depended on that guess. Solving for ∆T , and writing v instead of vi for simplicity, we find v2 2c ≈ 40◦ C ∆T = . The passengers would be boiled alive if not for the refrigeration.

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