Question

An aircraft travelling at $700\,km/h$ accelerating steadily at $10\,km/h\,{\text{per}}\,{\text{second}}$. Taking the speed of sound as $1100\,km/h$ at the aircraft, how long will it take to reach the sound barrier?

Answer 1

Hint-In order to find the time taken by the aircraft to cross the speed of sound we can use the first equation of motion which is given as
$v = u + at$
Where u is the initial velocity, v is the final velocity, t is the time taken, a is the acceleration.
By substituting the given values in this equation, we can find the time taken.

Step by step solution:
It is given that the velocity of an aircraft is $700\,km/h$ initially. So, we have
$u = 700\,km/h$
Let us convert this value given in kilometre per hour to meter per second. We know that
$1\,km = 1000\,m$ and $1\,h = 3600\,s$
Thus $\dfrac{{1\,km}}{{1\,h}} = \dfrac{{1000\,m}}{{3600\,s}} = \dfrac{{5\,m}}{{18\,s}}$
So to convert value given in $km/h$ to $m/s$ we need to multiply by the value by $\dfrac{5}{{18}}$
Therefore ,
$u = 700\, \times \dfrac{5}{{18}}\,m/s = \dfrac{{1750}}{9}\,m/s$
We need to find the time taken by the aircraft to cross the speed of sound . It is given that the speed of sound is $1100\,km/h$. This is the final velocity . so,
$v = 1100\,km/h$
$v = 1100\, \times \dfrac{5}{{18}}\,m/s = \dfrac{{2750}}{9}\,m/s$
Acceleration of the aircraft is $10\,km/h\,{\text{per}}\,{\text{second}}$ . that is
$a = 10\,km/h\,{\text{per}}\,{\text{second}}$
$a = 10\, \times \dfrac{5}{{18}}\,{\text{m/s/s = }}\dfrac{{25}}{9}\,m/{s^2}$
Now let us find the time taken to cross the speed of sound. For that we can use the first equation of motion which is given as
$v = u + at$
Where u is the initial velocity , v is the final velocity , t is the time taken , a is the acceleration.
Let us substitute the given values in this equation and solve for t .
$\dfrac{{2750}}{9} = \dfrac{{1750}}{9} + \dfrac{{25}}{9}\,t$
$ \Rightarrow 2750 = 1750 + 25t$
$ \Rightarrow 25t = 1000$
$\therefore t = 40\,s$
This is the time taken to cross the speed of sound.

Note: In this question the values of velocities are given in $km/h$ and the acceleration is given in $km/h/s$ don’t substitute these values directly in the first equation of motion. First convert all values to the standard units and then find the value of time.