Question

3. A stone is dropped in a well. (a) The sound of the splash is heat t seconds later. If d represents the depth of the well find an expression for d. (b) Find d when t 3

I would like some help with this question if you can! Thank you!

Answer 1

$t_1=\sqrt {\frac {2d}{g}}$ (time to hit the water)

$t_2=\frac {d }{v_{sound}}$ (time sound takes to travel the distance(depth))

$t=t_1+t_2$

$t=\sqrt {\frac {2d}{g}}+\frac {d}{v_{sound}}$

$v_{sound}t-d=v_{sound}\sqrt {\frac {2d}{g}}$

$v_{sound}^2t^2+d^2-2dv_{sound}t=\frac {2dv_{sound}^2}{g}$

$v_{sound}^2gt^2+gd^2-2gdv_{sound}t=2dv_{sound}^2$

$v_{sound}^2gt^2+gd^2-2gdv_{sound}t-2dv_{sound}^2=0$

$d^2-\left (2v_{sound}t+\frac {2v_{sound}^2}{g} \right )d+v_{sound}^2t^2=0$

$d=\frac {2v_{sound}t+\frac {2v_{sound}^2}{g}\pm \sqrt {\left ( 2v_{sound}t+\frac {2v_{sound}^2}{g} \right )^2-4v_{sound}^2t^2}}{2}$