Question

A ball is thrown straight up at speed $v_{0}$ in the presence of gravity.

(a) Ignoring air resistance at first, what is the maximum height the ball reaches?

(b) Now include Newtonian (quadratic) air resistance, with drag coefficient c. Letting the $\hat{y}$ direction point upwards, write down the equation of motion (be careful with the signs).

(c) Solve the equation of motion to find v(t)(= $\dot{y}$(t)). Remember to check that v(0) = $v_{0}$.

(d) Without integrating anything yet, calculate the time at which the ball reaches its maximum height.

(e) Find the formula y(t). (Note that $\frac{\mathrm{d} }{\mathrm{d} x}ln cos(x)=-tan(x)$ .)

(f) For a baseball thrown upwards at 45 mph (roughly 20 m/s), with g roughly 10 $m/s^{2}$ , mass m = 0.14 kg, and drag coefficient c = 1.4 × $10^{-3} Ns^{2}/m^{2}$ , calculate the height reached by the ball with and without air resistance.

Answer 1

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