Speed of Sound

Here I want to share a short method which can be used to find out the speed of the sound when you make a certain temperature half, double, thrice etc .


Let's start with an example.

Q: Temperature at which speed of sound becomes double as was at 27C is
a.273 C
b.0 C
c.927 C
d.1027 C

First of all see the traditional way of solving the problem which all follow:

Step 1: 
Conversion of centigrade to Kelvin:

                                           27 C = (27 + 273) K = 300K

Step 2:
Using the formula:

                                                  v₁ / v₂ = √T₁ / T₂

Squaring both sides we get :

                                                 (v₁/ v₂ )² = T₁/ T₂    --- (i)

Let v₁ is the speed at 27 C , and v₂ be the speed at new temperature, So according to the given data:
                                                       v₂ = 2v₁

using the value of v₂ in equation (i) we get:

                                                 (v₁ / 2v1 )² = T₁ / T₂   
this gives:

                                                   (1/2)² = T₁ / T₂   

this gives:
                                                      T₂ = 4 T₁

where T₁ is initial temperature which equals 300 K
 so we get,
                                                     T₂ = 1200K

Step 3:
Converting it back to Celsius we get

                                      1200 K = 1200 - 273 = 927 C

In all these steps what was actually done ?

Celsius is converted into Kelvin , which is then multiplied with some factor, in last step, and then converted back to Celsius.

In mathematical form we can write the above lines as:
                                              T₂ = k² ( T₁ + 273 )  - 273

Here k is the factor which depends how much change in the speed we want.
Remember in this formula T₁ is in Celsius , we are converting it in Kelvin in the brackets by adding 273...

Now applying the formula to given question:
here we have k=2 and T₁ =27 C

so T₂ = 4 ( 27 + 273 ) - 273 = 1200 - 273 = 927 and this is all.. Took hardly 30 sec to perform this calculation ..

Hope this will help you guys :)

In case of any confusion you can comment below..