question_answer

If \[{{\text{V}}_{\text{r}}}\] is the velocity of rain falling vertically and \[{{\text{V}}_{\text{m}}}\] is the velocity of a man walking on a level road, and \[\theta \] is the angle with vertical at which he should hold the umbrella to protect himself than the relative velocity of rain w.r.t. the man is given by:

A) \[{{\text{V}}_{\text{r}\,\text{m}}}=\sqrt{{{\text{V}}_{r}}^{2}+{{\text{V}}_{\text{m}}}^{2}+2{{\text{V}}_{r}}{{\text{V}}_{\text{m}}}\cos \theta }\]

B) \[{{\text{V}}_{\text{r}\,\text{m}}}=\sqrt{{{\text{V}}_{r}}^{2}+{{\text{V}}_{\text{m}}}^{2}-2{{\text{V}}_{r}}{{\text{V}}_{\text{m}}}\cos \theta }\]

C) \[{{\text{V}}_{\text{r}\,\text{m}}}=\sqrt{{{\text{V}}_{r}}^{2}+{{\text{V}}_{\text{m}}}^{2}}\]

D) \[{{\text{V}}_{\text{r}\,\text{m}}}=\sqrt{{{\text{V}}_{r}}^{2}-{{\text{V}}_{\text{m}}}^{2}}\]

Correct Answer: C

Solution :

[c] According to Pythagoras theorem \[{{V}_{r\,m}}=\sqrt{{{V}_{r}}^{2}+{{V}_{m}}^{2}}\]