Question

A man walking briskly in rain with speed $v$ must slant his umbrella forward making an angle with the vertical. A student derives the following relation between $\theta$ and $v$ : $\tan \theta = v$ and checks that the relation has a correct limit: as $v \to 0$ , $\theta \to 0$ , as expected. (We are assuming here is no strong wind and the rain falls vertically for a stationary man). Do you think this relation can be correct? If not, guess the correct relation.

Answer 1

Here we have to apply the process of dimensional analysis. We have to compare the dimension of both right hand side and left hand sides of the given equation, if found equal then the equation is correct otherwise wrong.
Dimensional Analysis (also known as Factor-Label Method or Unit Factor Method) is a problem-solving method that uses the fact that any number or expression can be multiplied by one without changing its value. It's a very effective strategy.

Complete step by step answer:
- Dimensions are the forces of which the fundamental quantities are expanded in order to represent those physical quantities..
- The major benefit of a problem dimensional analysis is that it reduces the number of variables in the problem by integrating dimensional variables to form non-dimensional parameters. By far the easiest and most desirable approach for analysing any fluid problem is the immediate mathematical solution.
- Quantities which are independent of all quantities are referred to as fundamental quantities. Units that are used to calculate these fundamental quantities are referred to as simple units.
- Quantities obtained using the original quantity shall be referred to as derived quantities. Units that are used to calculate these derivative quantities are referred to as derived units.
- Physical quantities having dimensions and a fixed value are called dimensional constants.
- Dimensional variables are real quantities that do not have dimensions and do not have a fixed value.
Given relation is:
$\tan \theta = v$
Dimension of R.H.S $= \left[ {{M^0}{L^1}{T^{ - 1}}} \right]$
Dimension of L.H.S $= \left[ {{M^0}{L^0}{T^0}} \right]$
Since $\tan \theta$ is a trigonometric function. So, it is a dimensionless quantity.
Since, the dimension of both sides is not equal. So, the relation is not correct.
To make the relation correct the dimensions of both sides should be the same. So, R.H.S should also be dimensionless.
We can do this by dividing the velocity with the velocity itself i.e. the velocity of rainfall.
$\tan \theta = \dfrac{v}{{v'}}$
Thus, both sides of the equation will be dimensionless. Hence, this relation is correct.

Note:
Here we have to know the correct dimensions of velocity and also we should know that the trigonometric quantities are dimensionless. The dimension of velocity is $= \left[ {{M^0}{L^1}{T^{ - 1}}} \right]$ and the dimension of $\tan \theta$ is $= \left[ {{M^0}{L^0}{T^0}} \right]$ .So, we have to be careful while solving this.
Dimensional formula is an equation in which the dimensions of the physical quantity are defined in terms of the fundamental quantity.