Intuitive Idea :
Assume that rain is falling at some constant velocity at some angle and our body's front area and top area are constant.Also there is shelter at some distance.
So,You're going to pass through a volume of watery air, and you're going to absorb all the water that's in the air that you pass through. The amount of water per unit volume doesn't change over time - the number of drops falling out of that volume are the same as the number falling in - so it doesn't matter when you get to a particular piece of that volume or how fast you're going when you get there. Imagine if we could stop gravity for a few minutes so the raindrops would just stay where they are. Would it matter how fast you went?Hence there will always be some constant amount of water you will face inspite of your velocity.
So where our velocity comes into picture?
Our velocity also has a role to play because rain has vertical velocity and horizontal velocity.So,we will get wet in top area due to the vertical velocity of rain as well as in front area due to horizontal velocity.
Recapitulating:Our wetness is due to three reasons of which one is constant and other two are due to rain's velocity.
Mathematical Approach: I am measuring the amount of wetness by the mass of water that falls on our body.
Let the density of water in surrounding be
$\rho$.
Front area of our body
$A$ and top area
$a$.
Suppose we are walking at velocity
v =
$v$ i.
And Rain's velocity is
$v_h$ i +
$v_v$ j.
{Ofcourse if angle from ground
$\theta$ is given then
$v_h$ =
$v_r\cos\theta$ and
$v_v$ =
$v_r\sin\theta$.}
Now, the Wetness
$$\begin{align*}W &= \text{mass of the water that fell on the body}\\ &= \text{density} \times \text{volume covered during the run}\\ &=\rho\ *\ [\underbrace{A(v+v_h)t}_\text{horizontal contribution} + \underbrace{av_vt}_\text{vertical contribution} ] \\ &=\rho\ *[\ Avt + av_vt + Av_ht]\\ &=\rho\ *[\ Ad + av_vt + Av_ht]\\ &=\rho\ *[\ Ad + av_v\frac{d}{v} + Av_h\frac{d}{v}]\\ \end{align*}$$ here
$d(=vt)$ is the nearest distance from the shelter.
Note Here I have taken the horizontal velocity of rain and our velocity in the opposite direction that's why the relative velocity is
$v+v_h$.
So, to minimize the wetness we have to increase
$v$ as it is in the denominator.
Also you can compare this relation with the
Intuitive Idea that I have given above about the
constant term and other terms arriving due to the horizontal and vertical velocity of rain.
If anybody finds any mistakes the please notify.