When the velocity v of an object is very large, the magnitude of the force due to air resistance is proportional to v2 with the force acting in opposition to the motion of the object. A shell of mass 3 kg is shot upward from the ground with an initial velocity of 500 m/sec. If the magnitude of the force due to air resistance is 0.1v2, when will the shell reach its maximum height above the ground? What is the maximum height?
Apply the formula for thermal speed
(m = 3, g = 9.81, b = 0.1)
When v = 500 m/sec and t = 0 then
The shell reaches at maximum height when v=0 and =17.15,
Hence, the shell reaches at the maximum height in 2.69 sec
When x=0 the value of c=-101.925.
Put t=2.69 then
Hence, the maximum height is 101.9248 m.
A parachutist whose mass is 75 kg drops from a helicopter hovering 2000 m above the ground and falls toward the ground under the influence of gravity. Assume that the force due to air resistance is proportional to the velocity of the parachutist, with the proportionality constant b1 = 30 N-sec/m when the chute is closed and b2= 90 N-sec/m when the chute is open. If the chute does not open until the velocity of the parachutist reaches 20 m/sec, after how many seconds will she reach the ground?
In Problem 14, suppose we have the additional information that the population of alligators on the grounds of the Kennedy Space Center in 1993 was estimated to be 4100. Use a logistic model to estimate the population of alligators in the year 2020. What is the predicted limiting population? [Hint: Use the formulas in Problem 12.
Two friends sit down to talk and enjoy a cup of coffee. When the coffee is served, the impatient friend immediately adds a teaspoon of cream to his coffee. The relaxed friend waits 5 min before adding a teaspoon of cream (which has been kept at a constant temperature). The two now begin to drink their coffee. Who has the hotter coffee? Assume that the cream is cooler than the air and has the same heat capacity per unit volume as the coffee, and that Newton’s law of cooling governs the heat transfer.
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