Question

A helicopter takes off along the vertical with an acceleration $a=3 \,m s^{-2}$ and zero initial velocity. In a certain time $t_{1}$, the pilot switches off the engine. At the point of take-off, the sound dies away in time $t_{2}=30 s$. Determine the velocity $v$ of the helicopter (in $m s^{-1}$ ) at the moment when its engine is switched off, assuming that the velocity $c$ of sound is $320 \,ms ^{-1}$.

Answer 1

$t_{2}=t_{1}+\frac{a t_{1}^{2}}{2 c}$
$t_{1}=\sqrt{\frac{c^{2}}{a}+\frac{2 c}{a} t_{2}}-\frac{c}{a}$
$v=a t_{1}=a \sqrt{\frac{c^{2}}{a}+\frac{2 c t_{2}}{a}}-\frac{c}{a}$
$=\sqrt{c^{2}+2 a c t_{2}}-c=80\, m s ^{-1}$