A vehicle starts from rest and accelerates along straight path at 2 m / s 2 . At the starting point of the vehicle, there is a stationary electric siren. How far has the vehicle nearly gone when the d

Question

A vehicle starts from rest and accelerates along straight path at 2 m / s 2 . At the starting point of the vehicle, there is a stationary electric siren. How far has the vehicle nearly gone when the driver hears the siren at 94% of its value when the vehicle was at rest? (speed of sound = 330 m / s )

TestHub · Accepted Answer

The apparent frequency heard by an observer moving away from a stationary source is given by the Doppler effect formula:

$f^{'} = f_{0} (\frac{v - v _{o}}{v})$

where $v$ is the speed of sound and $v_{o}$ is the speed of the observer (vehicle).

Given that the driver hears the siren at $94%$ of its original value, we have $f^{'} = 0.94 f_{0}$ . Substituting this and $v = 330 m/s$ into the formula:

$0.94 f_{0} = f_{0} (\frac{330 - v _{o}}{330})$

$0.94 = 1 - \frac{v _{o}}{330}$

$\frac{v _{o}}{330} = 1 - 0.94 = 0.06$

$v_{o} = 330 \times 0.06 = 19.8 m/s$

This is the velocity of the vehicle when the driver hears the sound. Now, using the kinematic equation $v^{2} = u^{2} + 2 a s$ for a vehicle starting from rest ( $u = 0$ ) with an acceleration $a = 2 m/s^{2}$ :

$(19.8)^{2} = 0^{2} + 2 (2) s$

$392.04 = 4 s$

$s = \frac{392.04}{4} = 98.01 m$

The vehicle has nearly gone $98 m$ .

Answer: 98 m

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