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A baseball is thrown vertically up with speed vo and is
Chapter 2, Problem 2.41(choose chapter or problem)
A baseball is thrown vertically up with speed \(v_{\mathrm{o}}\) and is subject to a quadratic drag with magnitude \(f(v)=c v^{2}\). Write down the equation of motion for the upward journey (measuring y vertically up) and show that it can be rewritten as \(\dot{v}=-g\left[1+\left(v / v_{\text {ter }}\right)^{2}\right]\). Use the "v dv dx rule"
(2.86) to write \(\dot{v}\) as v dv/dy, and then solve the equation of motion by separating variables (put all terms involving v on one side and all terms involving y on the other). Integrate both sides to give y in terms of v, and hence v as a function of y. Show that the baseball's maximum height is
\(y_{\max }=\frac{v_{\text {ter }}^{2}}{2 g} \ln \left(\frac{v_{\text {ter }}^{2}+v_{\mathrm{o}}^{2}}{v_{\text {ter }}^{2}}\right) .\)
If \(v_{\mathrm{o}}=20 \mathrm{~m} / \mathrm{s}\) (about 45 mph) and the baseball has the parameters given in Example 2.5 (page 61), what is \(y_{\max }\)? Compare with the value in a vacuum.
Questions & Answers
QUESTION:
A baseball is thrown vertically up with speed \(v_{\mathrm{o}}\) and is subject to a quadratic drag with magnitude \(f(v)=c v^{2}\). Write down the equation of motion for the upward journey (measuring y vertically up) and show that it can be rewritten as \(\dot{v}=-g\left[1+\left(v / v_{\text {ter }}\right)^{2}\right]\). Use the "v dv dx rule"
(2.86) to write \(\dot{v}\) as v dv/dy, and then solve the equation of motion by separating variables (put all terms involving v on one side and all terms involving y on the other). Integrate both sides to give y in terms of v, and hence v as a function of y. Show that the baseball's maximum height is
\(y_{\max }=\frac{v_{\text {ter }}^{2}}{2 g} \ln \left(\frac{v_{\text {ter }}^{2}+v_{\mathrm{o}}^{2}}{v_{\text {ter }}^{2}}\right) .\)
If \(v_{\mathrm{o}}=20 \mathrm{~m} / \mathrm{s}\) (about 45 mph) and the baseball has the parameters given in Example 2.5 (page 61), what is \(y_{\max }\)? Compare with the value in a vacuum.
ANSWER:Step 1 of 7
Two forces act on a ball downward: a drag force and a gravity force when the ball is projected upwards.
The equation of motion for the ball can be written as,
\(\dot{m v}=-m g-c v^{2}\)
Here, mg is the weight acting downward and \(c v^{2}\) is a drag force also acting downward.