Question

You throw a ball vertically at a height H. What height does the ball have to reach for it to spend twice as long in the air? (Note: you need to get an equation)

Answer 1

$Suppose\,\, the \,\,ball\,\, was \,\,thrown \,\,at \,\,a\,\, velocity \,\,v_i\,\,then\\ v_f^2=v_i^2-2gh\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,where\,\,v_f \,\,is\,\,final\,\,velocity\,\,and\,\,h\,\,is\,\,height.\\$
  $g\,\, is\,\,the\,\, acceleration\,\,due\,\,to\,\,gravity.$

$Now\,\,at\,\,the\,\,maximum\,\,height\,\,h=H\,\,and\,\,v_f=0\\ \therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0=v_i^2-2gH \\\Rightarrow \,\,\,\,v_i=\sqrt{2gH}$

$When\,\,the\,\,ball\,\,will\,\,come\,\,down\,\,from\,\,height\,\,H \\ \therefore \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,v_f^2=0+2gH\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,initial\,\,velocity\,\,(v_i)\,\,will\,\,be\,\,0 \\\Rightarrow \,\,\,\,v_f=\sqrt{2gH}$

$If\,\,the\,\,time\,\,taken\,\,to\,\,reach\,\,height\,\,H\,\,is\,\,t_1\,\,then,\\ 0=\sqrt{2gH}-gt_1 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\[from\,\,\, v_f=v_i-gt]\\\\ t_1=\sqrt{\frac{2H}{g}}$

$If\,\,the\,\,time\,\,taken\,\,to\,\,fall\,\,down\,\,from\,\,height\,\,H\,\,is\,\,t_2\,\,then,\\ \sqrt{2gH}=0+gt_2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\[from\,\,\, v_f=v_i+gt]\\\\ t_2=\sqrt{\frac{2H}{g}}$

$So\,\,the\,\,ball\,\,takes\,\,same\,\,time\,\,to\,\,reach\,\,a\,\,height\,\,H\,\,and\,\,to\,\,fall\,\,down\,\,from\,\,the\,\,same\,\,height.$
$Therefore\,\,each\,\,time\,\,will\,\,be\,\,twice\,\,to\,\,spend\,\,twice\,\,as\,\,long\,\,in\,\,the\,\,air.\\ If\,\,new\,\,initial\,\,velocity\,\,be\,\,v_i\,\,then,\\ 0=v_i-g\left (2\,\,\sqrt{\frac{2H}{g}} \right )\,\,\,\,\,\,\,\,\,\,\,\,\,\,[from\,\,\,\,v_f=v_i-gt]\\ v_i=\sqrt{8gH}\\$

$Now\,\,suppose\,\,will\,\,reach\,\,a\,\,height\,\,h_2\,\,spending\,\,twice\,\,as\,\,long\,\,in\,\,the\,\,air.\\ \\ 0=v_i^2-2gh_2\\ 0=\left (\sqrt{8gH} \right )^2-2gh_2\\ 2gh_2=8gH\\ h_2=4H$

$\dpi{120} \textbf{The\,\,ball\,\,have\,\,to\,\,reach\,\,a\,\,height\,\,4H\,\,to\,\,spend\,\,twice\,\,as\,\,long\,\,in\,\,the\,\,air.}$