Question

A parachutist jumps from the top of a very high tower with a siren of frequency 800 Hz on his back. Assume the initial velocity to be zero. After falling freely for 12s, he observes that the frequency of sound heard by him reflected from level ground below him is differing by 700 Hz w.r.t. the original frequency. What was the height of the tower. Velocity of sound in air is 330 m/s and $g=10m/{{s}^{2}}$
A . 511.5 m
B . 1057.5 m
C . 757.5 m
D . 1215.5 m

Answer 1

Hint: In this question, we have to find the height of the tower. For this, we use the relation between apparent frequency and the original frequency. By putting the values, we get an equation in the form of a quadratic equation and solving the equations, we get the value of the height of the tower.

Formula Used:
The formula of apparent frequency is,
$f'=f\left( \dfrac{v+g{{t}_{0}}}{v-g{{t}_{1}}} \right)$
Where, $g{{t}_{1}}$ is the velocity of source at the instant of sound, $g{{t}_{0}}$ is the velocity of observer at the instant of observing some sound and $f$ is the original frequency.

Complete step by step solution:
Let the sound observed by a parachutist at ${{t}_{0}}=12s$ be produced at ${{t}_{1}}s$.
Velocity of source at the instant of sound = $g{{t}_{1}}$
And the velocity of observer at the instant of observing some sound = $g{{t}_{0}}$
Hence we know the relation between apparent frequency $f'$ and the original frequency f will be,
$f'=f\left( \dfrac{v+g{{t}_{0}}}{v-g{{t}_{1}}} \right)$
Here $f=800Hz$, $g=10m/{{s}^{2}}$, $v=330m/s$ and ${{t}_{0}}=12s$
And $f'=800+700=1500\,Hz$

Now, after putting these value in the expression of apparent frequency is,
$1500=800\left( \dfrac{330+10(12)}{330-10({{t}_{1}})} \right) \\ $
Solving the above equation, we get
$330-10({{t}_{1}})=\dfrac{450\times 800}{1500} \\ $
Then $10{{t}_{1}}=90$
And ${{t}_{1}}=9s$
Now the distance travelled by sound in $({{t}_{0}}-{{t}_{1}})$ sec is,
$v({{t}_{0}}-{{t}_{1}})=\left( h-\dfrac{1}{2}g{{t}_{0}}^{2} \right)+\left( h-\dfrac{1}{2}g{{t}_{1}}^{2} \right)$

Now we put all the values in the above equation, we get
$330(12-9)=\left( h-\dfrac{1}{2}(10){{(12)}^{2}} \right)+\left( h-\dfrac{1}{2}(10){{(9)}^{2}} \right) \\ $
On simplifying the above equation, we get
$990=\left( h-720 \right)+\left( h-405 \right) \\ $
It forms a quadratic equation which is ${{h}^{2}}-1125h-290610=0$.
We solve the above equation with the help of a quadratic formula.
By solving it, we get the value of $h=1057.5\,m$.
Hence, the height of the tower is $1057.5\,m$.

Hence, option B is correct.

Note: In this question, It is important to remember the relation to solve this question Otherwise first we have to derive the relation then we are able to find the height of the tower. Students must take care while solving these equations.