Chapter 2.10, Problem 229RP

To determine

The initial velocity which the mass center of player possess.

Answer to Problem 229RP

The initial velocity which the mass center of player possess is $13.9\text{ft}/\text{s}$.

Explanation of Solution

Write the expression of acceleration that acts vertically downward direction opposite to the direction of motion.

$a=-g$ (I)

Here, acceleration due to gravity is g.

Conclusion:

Substitute $\left(v\frac{dv}{dh}\right)$ for a in Equation (I).

$v\frac{dv}{dh}=-g$

$vdv=-gdh$

Integrate the equation.

${\displaystyle \underset{v_0}{\overset{0}{\int }}vdv}=-{\displaystyle \underset{h}{\overset{h+3}{\int }}gdh}$

$\begin{array}{l}{\left[\frac{v^2}{2}\right]}_{v_0}^0=-g{\left[h\right]}_h^{h+3}\\ \left(0-\frac{v_0^2}{2}\right)=-g\left[\left(h+3\right)-h\right]\\ -\frac{v_0^2}{2}=-3g\end{array}$

$v_0=\sqrt{6g}$ (II)

Substitute $32.2\text{ft}/{\text{s}}^2$ for g in Equation (II).

$\begin{array}{c}{v}_0=\sqrt{6\times 32.2\text{ft}/{\text{s}}^2}\\ =\sqrt{193.2}\\ =13.9\text{ft}/\text{s}\end{array}$

Thus, the initial velocity which the mass center of player possess is $13.9\text{ft}/\text{s}$.