# A Sound Correction

As I was taking my coffee over breakfast today, I realized that in the previous post, I did not incorporate the time it took for the sound waves to reach my ear when I pressed the stop button of the stopwatch. Sound waves travel at 340.29 meters per second. At a height of 25 meters, it would take 0.073 seconds for the sound of the explosion to reach the ground. This should be negligible for the purposes of our computation. However, what is really the effect if we incorporate this minor correction?

As seen in the figure below, the recorded time is composed of the time it took for the skyrocket to reach the maximum height plus the time it took for the sound waves to reach my ear, that is,

$t_r = t_a + t_s$

where $t_r$ is the recorded time, $t_a$ is the actual time the skyrocket reached its maximum height and $t_s$ is the time it took for the sound waves to travel from the maximum height to the ground.

We can rewrite the height as

$\displaystyle h_{\text{max}} = - \frac{gt_a^2}{2} + v_i t_a$

Since $v_i = gt_a$,

$\displaystyle h_{\text{max}} = - \frac{gt_a^2}{2} + gt_a^2$
$\displaystyle = \frac{gt_a^2}{2}$

At the maximum height, $h_{\text{max}}$, it would take the sound waves $t_s$ seconds to travel, that is,

$\displaystyle h_{\text{max}} = v_s t_s$

where $v_s$ is the speed of sound. Therefore,

$\displaystyle t_sv_s = \frac{gt_a^2}{2}$
$\displaystyle (t_r- t_a)v_s = \frac{gt_a^2}{2}$
$\displaystyle v_st_r - v_st_a = \frac{gt_a^2}{2}$
$\displaystyle \frac{gt_a^2}{2} + v_st_a - v_st_r = 0$

The last equation above is a quadratic equation which we can solve for $t_a$ using the quadratic formula:

$\displaystyle t_a = \frac{-v_s \pm \sqrt{v_s^2 - 4(1/2)gv_st_r}}{2(1/2)g}$

Substituting the values $g = 9.8$, $v_s=340.29$, and $t_r = 2.3$, we get

$\displaystyle t_a = 2.22$

Using this result, the maximum height is computed to be

$\displaystyle h_{\text{max}} = \frac{gt_a^2}{2} = 24.33$

Comparing this with our previous computation of $h_{\text{max}} = 25.829$, we find that we have overestimated the maximum height by about 1.49 meters. It’s not really that bad so we can just drop the effect of sound.