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[tex]\textrm{Hello, folks. I just want to check my work on this problem. Thanks.}[/tex]
[tex]\textrm{A certain ball has the property that each time it falls from a height}[/tex] [tex]h[/tex] [tex]\textrm{onto a hard, level surface, it rebounds to a height}[/tex] [tex]rh[/tex] [tex]\textrm{, where}[/tex] [tex]0<r<1[/tex]. [tex]\textrm{Suppose that the ball is dropped from an initial height of}[/tex] [tex]H[/tex] [tex]\textrm{meters.}[/tex]
[tex]\textrm{(a) Assuming that the ball continues to bounce indefinitely, find the total distance that<br /> it travels.}[/tex]
[tex]H + 2rH + 2r^{2}H + 2r^{3}H + \cdots = H + 2H \sum _{n=1} ^{\infty} \left( r \right) r^{n-1} = H + 2H \left( \frac{r}{1-r} \right) = H \left( \frac{1+r}{1-r} \right)[/tex]
[tex]\textrm{(b) Calculate the total time that the ball travels.}[/tex]
[tex]t_{\textrm{TOTAL}} = \sqrt{\frac{2H}{g}} + \sqrt{\frac{2rH}{g}} + \sqrt{\frac{2r^2H}{g}} + \sqrt{\frac{2r^3 H}{g}} + \cdots[/tex]
[tex]t_{\textrm{TOTAL}} = \sqrt{\frac{2H}{g}} + \sqrt{\frac{2rH}{g}} \left( 1 + \sqrt{r} + \sqrt{r^2} + \sqrt{r^3} + \cdots \right)[/tex]
[tex]t_{\textrm{TOTAL}} = \sqrt{\frac{2H}{g}} + \sqrt{\frac{2rH}{g}} \left( \frac{1}{1-\sqrt{r}} \right)[/tex]
[tex]\textrm{(c) Suppose that that each time the ball strikes the surface with velocity}[/tex] [tex]v[/tex] [tex]\textrm{it rebounds with velocity}[/tex] [tex]-kv[/tex][tex]\textrm{, where}[/tex] [tex]0<k<1[/tex]. [tex]\textrm{How long will it take for the ball to come <br /> to rest?}[/tex]
[tex]v_{\textrm{REST}} = v + kv + k^2 v + k^3 v + \cdots[/tex]
[tex]v_{\textrm{REST}} = v + \sum _{n=1} ^{\infty} \left( k v \right) k^{n-1}[/tex]
[tex]v_{\textrm{REST}} = v + \left( \frac{kv}{1-k} \right)[/tex]
[tex]\textrm{If } K=U, \textrm{we find}[/tex]
[tex]\frac{1}{2}mv_{\textrm{REST}} ^2= mgH[/tex]
[tex]\frac{1}{2}m\left[ v^2 + 2v^2 \left( \frac{k}{1-k} \right) + v^2 \left( \frac{k}{1-k} \right)^2 \right] = mgH[/tex]
[tex]H = \frac{1}{2g}\left[ v^2 + 2v^2 \left( \frac{k}{1-k} \right) + v^2 \left( \frac{k}{1-k} \right)^2 \right][/tex]
[tex]\textrm{which gives}[/tex]
[tex]t_{\textrm{REST}} = - \frac{2\left| \frac{v}{g\left( k-1 \right)} \right|}{\sqrt{r}-1} - \left| \frac{v}{g\left( k-1 \right)} \right|[/tex]
[tex]\textrm{A certain ball has the property that each time it falls from a height}[/tex] [tex]h[/tex] [tex]\textrm{onto a hard, level surface, it rebounds to a height}[/tex] [tex]rh[/tex] [tex]\textrm{, where}[/tex] [tex]0<r<1[/tex]. [tex]\textrm{Suppose that the ball is dropped from an initial height of}[/tex] [tex]H[/tex] [tex]\textrm{meters.}[/tex]
[tex]\textrm{(a) Assuming that the ball continues to bounce indefinitely, find the total distance that<br /> it travels.}[/tex]
[tex]H + 2rH + 2r^{2}H + 2r^{3}H + \cdots = H + 2H \sum _{n=1} ^{\infty} \left( r \right) r^{n-1} = H + 2H \left( \frac{r}{1-r} \right) = H \left( \frac{1+r}{1-r} \right)[/tex]
[tex]\textrm{(b) Calculate the total time that the ball travels.}[/tex]
[tex]t_{\textrm{TOTAL}} = \sqrt{\frac{2H}{g}} + \sqrt{\frac{2rH}{g}} + \sqrt{\frac{2r^2H}{g}} + \sqrt{\frac{2r^3 H}{g}} + \cdots[/tex]
[tex]t_{\textrm{TOTAL}} = \sqrt{\frac{2H}{g}} + \sqrt{\frac{2rH}{g}} \left( 1 + \sqrt{r} + \sqrt{r^2} + \sqrt{r^3} + \cdots \right)[/tex]
[tex]t_{\textrm{TOTAL}} = \sqrt{\frac{2H}{g}} + \sqrt{\frac{2rH}{g}} \left( \frac{1}{1-\sqrt{r}} \right)[/tex]
[tex]\textrm{(c) Suppose that that each time the ball strikes the surface with velocity}[/tex] [tex]v[/tex] [tex]\textrm{it rebounds with velocity}[/tex] [tex]-kv[/tex][tex]\textrm{, where}[/tex] [tex]0<k<1[/tex]. [tex]\textrm{How long will it take for the ball to come <br /> to rest?}[/tex]
[tex]v_{\textrm{REST}} = v + kv + k^2 v + k^3 v + \cdots[/tex]
[tex]v_{\textrm{REST}} = v + \sum _{n=1} ^{\infty} \left( k v \right) k^{n-1}[/tex]
[tex]v_{\textrm{REST}} = v + \left( \frac{kv}{1-k} \right)[/tex]
[tex]\textrm{If } K=U, \textrm{we find}[/tex]
[tex]\frac{1}{2}mv_{\textrm{REST}} ^2= mgH[/tex]
[tex]\frac{1}{2}m\left[ v^2 + 2v^2 \left( \frac{k}{1-k} \right) + v^2 \left( \frac{k}{1-k} \right)^2 \right] = mgH[/tex]
[tex]H = \frac{1}{2g}\left[ v^2 + 2v^2 \left( \frac{k}{1-k} \right) + v^2 \left( \frac{k}{1-k} \right)^2 \right][/tex]
[tex]\textrm{which gives}[/tex]
[tex]t_{\textrm{REST}} = - \frac{2\left| \frac{v}{g\left( k-1 \right)} \right|}{\sqrt{r}-1} - \left| \frac{v}{g\left( k-1 \right)} \right|[/tex]