Final answer:
The problem requires knowledge of relative velocity and vector components to determine the speed of raindrops with respect to the road and the moving man. The man's running speed is equal to the horizontal velocity of the rain, and the vertical component of the rain's speed is the speed relative to the man when he starts running.
Step-by-step explanation:
The question involves applying principles of relative velocity to determine the speed of raindrops with respect to the road and a moving man. When someone is under rain that's falling vertically but has to hold an umbrella at an angle to stay dry, the need for this angle suggests a horizontal component to the rain's velocity.
Here, the man initially perceives the rain to be falling at an angle, which leads to the conclusion that raindrops have a horizontal velocity component.
Given that the man has to start running to remove the horizontal component and perceive the raindrops as falling vertically, his running speed becomes equal to the horizontal component of the rain's velocity. Thus, we can equate the man's running speed to the horizontal velocity of raindrops with respect to the road.
The speed of the raindrops relative to the man would then be only the vertical component because the horizontal component is canceled by the man's running speed.
To find the numerical answers, you would need the actual vertical velocity of the raindrops and the angle at which the man holds his umbrella. Unfortunately, the numerical data isn't provided in this question.
However, with those values, you could use trigonometric relationships to solve for the horizontal speed of the rain and combine it with the vertical speed to get the total speed with respect to the road. For the speed with respect to the man, only the vertical component would be considered.