Science:Math Exam Resources/Courses/MATH102/December 2019/Question 11

MATH102 December 2019

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Question 11
A ball is thrown in the air so that initially it is travelling upwards(the positive direction) at 10m/sec. It accelerates under gravity and air resistance, so that its vertical velocity v(t) satisfies the differential equation $v'(t)=-10-v(t)$ (here t is measured in seconds and v is measured in m/sec). How many seconds will it take for the ball reach the peak of its arc?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Apply separation of variables to solve the first order differential equation.

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Solution

Whoops, no solution has been written yet. Be the first to submit a suggestion. We use separation of variables to solving the first order differential equation ${\begin{aligned}{\frac {dv}{dt}}&=-10-v\\{\frac {dv}{10+v}}&=-dt\\v(t)&=ce^{-t}-10\\\end{aligned}}$ Since the initial velocity is v(0) = 10, we have the constant c = 20. When it reaches the peak of its arc, the velocity is 0. Thus solving for $v(t)=20e^{-t}-10=0,\implies t=-\ln({\frac {1}{2}})$