Answer
Verified
432.3k+ views
Hint:We use here the concept of relative velocity in two dimensions as we know if we want to find velocity of rain with respect to man then we have to subtract velocity of man from the velocity of rain with respect to ground. By using this relation we can calculate velocity of rain with respect to ground.
Step by step solution:
Let’s assume the velocity of rain with respect to ground is ${V_{RG}}$ which is $30^\circ $ with the vertical and velocity of man with respect to ground is given ${V_{MG}} = 10kmph$ and velocity of rain with respect to man ${V_{RM}}$ is given it is perpendicular to man
Velocity of rain with respect to ground is given $30^\circ $with the vertical, horizontal and vertical component are shown in figure
Horizontal component of velocity of rain with respect to ground is ${V_{RG}}\cos 60 = \dfrac{1}{2}{V_{RG}}$
And the vertical component is ${V_{RG}}\sin 60 = \dfrac{{\sqrt 3 }}{2}{V_{RG}}$
So we can write the velocity of rain with respect to ground in vector form $ \Rightarrow {\vec V_{RG}} = \dfrac{1}{2}{V_{RG}}\hat i - \dfrac{{\sqrt 3 }}{2}{V_{RG}}\hat j$
And velocity of man with respect to ground is ${\vec V_{MG}} = 10\hat i$
Now the velocity of rain with respect to man can find as
$ \Rightarrow {\vec V_{RM}} = {\vec V_{RG}} - {\vec V_{MG}}$
Put the value of ${\vec V_{RG}}$ and ${\vec V_{MG}}$ from above
\[ \Rightarrow {\vec V_{RM}} = \dfrac{1}{2}{V_{RG}}\hat i - \dfrac{{\sqrt 3 }}{2}{V_{RG}}\hat j - 10\hat i\]
Rearranging
\[ \Rightarrow {\vec V_{RM}} = \left( {\dfrac{1}{2}{V_{RG}} - 10} \right)\hat i - \dfrac{{\sqrt 3 }}{2}{V_{RG}}\hat j\]....................... (1)
Now in the question it is given that He finds that rain drops are hitting his head vertically its means the velocity of rain with respect to man have only vertical component $( - \hat j)$
So the horizontal component must be zero in above equation means coefficient of $\hat i$ must be zero
\[ \Rightarrow \left( {\dfrac{1}{2}{V_{RG}} - 10} \right) = 0\]
\[ \Rightarrow \dfrac{1}{2}{V_{RG}} = 10\]
Solving it
$\therefore {V_{RG}} = 20$
So we get the magnitude of velocity of rain with respect to ground is $20kmph$
Hence option B is correct
Note:By this simple concept we can solve these types of question easily in this question velocity of rain with respect to man is also asked then we can solve it in simple manner we just put the value of ${V_{RG}}$ in equation (1) and will get the answer.
Step by step solution:
Let’s assume the velocity of rain with respect to ground is ${V_{RG}}$ which is $30^\circ $ with the vertical and velocity of man with respect to ground is given ${V_{MG}} = 10kmph$ and velocity of rain with respect to man ${V_{RM}}$ is given it is perpendicular to man
Velocity of rain with respect to ground is given $30^\circ $with the vertical, horizontal and vertical component are shown in figure
Horizontal component of velocity of rain with respect to ground is ${V_{RG}}\cos 60 = \dfrac{1}{2}{V_{RG}}$
And the vertical component is ${V_{RG}}\sin 60 = \dfrac{{\sqrt 3 }}{2}{V_{RG}}$
So we can write the velocity of rain with respect to ground in vector form $ \Rightarrow {\vec V_{RG}} = \dfrac{1}{2}{V_{RG}}\hat i - \dfrac{{\sqrt 3 }}{2}{V_{RG}}\hat j$
And velocity of man with respect to ground is ${\vec V_{MG}} = 10\hat i$
Now the velocity of rain with respect to man can find as
$ \Rightarrow {\vec V_{RM}} = {\vec V_{RG}} - {\vec V_{MG}}$
Put the value of ${\vec V_{RG}}$ and ${\vec V_{MG}}$ from above
\[ \Rightarrow {\vec V_{RM}} = \dfrac{1}{2}{V_{RG}}\hat i - \dfrac{{\sqrt 3 }}{2}{V_{RG}}\hat j - 10\hat i\]
Rearranging
\[ \Rightarrow {\vec V_{RM}} = \left( {\dfrac{1}{2}{V_{RG}} - 10} \right)\hat i - \dfrac{{\sqrt 3 }}{2}{V_{RG}}\hat j\]....................... (1)
Now in the question it is given that He finds that rain drops are hitting his head vertically its means the velocity of rain with respect to man have only vertical component $( - \hat j)$
So the horizontal component must be zero in above equation means coefficient of $\hat i$ must be zero
\[ \Rightarrow \left( {\dfrac{1}{2}{V_{RG}} - 10} \right) = 0\]
\[ \Rightarrow \dfrac{1}{2}{V_{RG}} = 10\]
Solving it
$\therefore {V_{RG}} = 20$
So we get the magnitude of velocity of rain with respect to ground is $20kmph$
Hence option B is correct
Note:By this simple concept we can solve these types of question easily in this question velocity of rain with respect to man is also asked then we can solve it in simple manner we just put the value of ${V_{RG}}$ in equation (1) and will get the answer.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Which are the Top 10 Largest Countries of the World?
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Write a letter to the principal requesting him to grant class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE