Question

When sound is produced in an aeroplane moving with the velocity of $200m/s$ horizontal, its echo is heard after $10\sqrt{5}s$. If the velocity of sound in air is $300m/s$, the elevation of the aircraft is,

• A. $250m$
• B. $250\sqrt{5}m$
• C. $12.5m$
• D. $2500m$

Answer 1

The correct option is D

$2500m$

Step 1. Given Data:

The velocities are given. Horizontal velocity of the aeroplane, $v_1=200m/sand{v}_2=300m/s$

Where $v_2$ is the velocity of sound in air. And their corresponding time $t_1=10\sqrt{5}sand{t}_2=5\sqrt{5}s$

Step 2. Formula used:

Using the equation of velocity as, $dis\tan ce=velocity\times time$

Step 3. Applying the formula:

The velocity $v_1$ is horizontal and hence the distance traveled by it ($D_1$) is horizontal only whereas $v_2$ is the echo heard which depends on the velocity of sound in air and its distance trtraveleds $D_2$.

The equation of velocity is $dis\tan ce=velocity\times time$

$\begin{array}{rcl}{D}_1& =& v_1{t}_1\\ & =& 200\times 5\sqrt{5}\\ & =& 2236m\\ D_2& =& v_2{t}_2\\ & =& 300\times 5\sqrt{5}\\ & =& 3354m\end{array}$

When the distances are measured, they form a right-angled triangle with elevation height has one of its sides denoted by h.

$$\begin{gathered} D_2^2=D_1^2+h^2 \\ {h}^2={\left(3354\right)}^2-{\left(2236\right)}^2 \\ h=2500m \end{gathered}$$

Hence, the elevation of the airplane is $2500m.$

Therefore, Option D is the correct answer.