A helicopter takes off along the vertical with an acceleration a=3 m s-2 and zero initial velocity. In a certain time t1, the pilot switches off the engine. At the point of take-off, the sound dies away in time t2=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.

Question

A helicopter takes off along the vertical with an acceleration a=3 m s-2 and zero initial velocity. In a certain time t1, the pilot switches off the engine. At the point of take-off, the sound dies away in time t2=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. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

t2=t1+(a t12/2 c) t1=√(c2/a)+(2 c/a) t2-(c/a) v=a t1=a √(c2/a)+(2 c t2/a)-(c/a) =√c2+2 a c t2-c=80 m s -1

t2​=t1​+2cat12​​ t1​=ac2​+a2c​t2​​−ac​ v=at1​=aac2​+a2ct2​​​−ac​ =c2+2act2​​−c=80ms−1

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