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An object with mass m is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If s(t) is the distance dropped after t seconds, then the speed is v = s'(t) and the acceleration is a = v'(t). If g is the acceleration due to gravity, then the downward force on the object is mg - cv, where c is a positive constant, and Newton's Second Law gives mdv mg - cv. dt (a) Solve this as a linear equation. (Use v for v(t).) (b) What is the limiting velocity? lim v(t) = (c) Find the distance the object has fallen after t seconds. (Use s for s(t).)

An object with mass m is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If s(t) is the distance dropped after t seconds, then the speed is v = s'(t) and the acceleration is a = v'(t). If g is the acceleration due to gravity, then the downward force on the object is mg - cv, where c is a positive constant, and Newton's Second Law gives mdv mg - cv. dt (a) Solve this as a linear equation. (Use v for v(t).) (b) What is the limiting velocity? lim v(t) = (c) Find the distance the object has fallen after t seconds. (Use s for s(t).)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 18EQ
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An object with mass m is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If s(t) is the distance dropped after t seconds, then the speed is v = s'(t) and the acceleration is a = v'(t). If g is the acceleration due to gravity, then the downward force on the object is mg – cv, where c is a positive constant, and Newton's Second Law gives dv m- mg – cv. - dt (a) Solve this as a linear equation. (Use v for v(t).) (b) What is the limiting velocity? lim v(t) t → ∞ (c) Find the distance the object has fallen after t seconds. (Use s for s(t).)
Transcribed Image Text:An object with mass m is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If s(t) is the distance dropped after t seconds, then the speed is v = s'(t) and the acceleration is a = v'(t). If g is the acceleration due to gravity, then the downward force on the object is mg – cv, where c is a positive constant, and Newton's Second Law gives dv m- mg – cv. - dt (a) Solve this as a linear equation. (Use v for v(t).) (b) What is the limiting velocity? lim v(t) t → ∞ (c) Find the distance the object has fallen after t seconds. (Use s for s(t).)
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