AP Calculus BC · Integration and accumulation of change
Take downward as the positive direction
Problem
Take downward as the positive direction. A body of mass \(m=1\) (in consistent units) is released from rest and obeys \[ \frac{dv}{dt}=9-v^{2},\qquad v(0)=0, \] where \(v(t)\) is the downward speed. Let \(s(t)\) be the downward distance traveled, with \(s(0)=0\). (a) Identify the terminal speed \(v_{\infty}=\displaystyle\lim_{t\to\infty}v(t)\) from the differential equation (without solving it). Then separate variables and solve for \(v(t)\) explicitly, writing the answer in exponential form. (b) Integrate \(v(t)\) to obtain \(s(t)\). (c) Find the exact time and the exact distance at which the body first reaches \(90\%\) of its terminal speed. Confirm that every logarithm appearing in the work has a positive argument.
Hint
Terminal speed is the positive root of \(9-v^{2}=0\); the IVP then separates.
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