Let vr= \(a\hat i+b\hat j \)
Velocity of girl =vg = (5 m/s)\(\hat i\)
Velocity of rain w.r.t. girl
vrg=vr−vg= \((a\hat i+b\hat j)-5\hat i\)
= (a – 5)\(\hat i\)+ b\(\hat j\)
Hence, a – 5= 0 ⇒ a = 5
Now, vg= (10\(\frac{m}{s}\) )\(\hat i\)
vrg= vr−vg
= \((a\hat i+b\hat j)-10\hat i\)
= (a – 10)\(\hat i+b \hat j\)
As, angle appear 45º,
∴ tan 45º =\(\frac{b}{a-10}\)= 1
⇒b = a – 10 = 5 – 10 = −5
Hence, velocity of rain = \(a\hat i+b\hat j\)
⇒vr = \(5\hat i-5\hat j\)
Speed of rain
= \(\begin{vmatrix} v_r\end {vmatrix}\) = \(\sqrt{(5)^2+(-5)^2}\)
= \(\sqrt50=5\sqrt2\,m/s\).