A girl riding a bicycle with a speed of 5ms−1 towards north direction, observes rain falling vertically down. If she increases her speed to 10ms−1, rain appears to meet her at 45° to the vertical. What is the speed of the rain?

Assume north to be i^ direction and vertically upwards to be j^ direction.Let the velocity of rain be v→r=ai^+bj^In the first case, v→g= velocity of girl=5i^∴v→rg=v→r−v→g=(ai^+bj^)−5i^=(a−5)i^+bj^Since rain appears to fall vertically downwards,∴ a−5=0 or a=5In the second case, v→g=10i^∴ v→rg=v→r−10i^=(ai^+bj^)−10i^=(a−10)i^+bj^=−5i^+bj^Since rain appears to fall at 45° with the vertical, ∴ a=b=−5∴ The velocity of the rain is v→r=−5i^−5j^Speed of the rain is v→r=(−5)2+(−5)2=52ms−1