A girl standing on road holds her umbrella at 45° with the vertical to keep the rain away. If she starts running without umbrella with a speed of 1 5 2 k m h − 1 , the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is:

Let V R G r (velocity of Rain w.r.t. ground or w.r.t to standing Girl) V → R G r = V x i ^ + V y j ^ t a n ( 2 7 0 + 4 5 ° ) = − c o t 4 5 ° = V y V x = − 1 Now, velocity of Rain w.r.t. moving Girl V → R G m = ( V x − 1 5 2 ) i ^ + V y j ^ = ( 1 5 2 − 1 5 2 ) i ^ + V y j ^ = V y j ^ V → R G m = − 1 5 2 j ^ ⇒ V → R G m = − 3 0 2 j ^ ⇒ | V → R G m | = 3 0 2

A girl standing on road holds her umbrella at 45° with the vertical to keep the rain away.

If she starts running without umbrella with a speed of $15 \sqrt[]{2} k m h^{- 1},$ the rain drops hit her head vertically. The speed of rain drops with respect to the moving girl is:

Option 1 - 30 kmh-1

Option 2 - <math> <mrow> <mfrac> <mrow> <mn>2</mn> <mn>5</mn> </mrow> <mrow> <mroot> <mrow> <mn>2</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> <mi>k</mi> <mi>m</mi> <msup> <mrow> <mi>h</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math>

Option 3 - <math> <mrow> <mfrac> <mrow> <mn>3</mn> <mn>0</mn> </mrow> <mrow> <mroot> <mrow> <mn>2</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> <mi>k</mi> <mi>m</mi> <msup> <mrow> <mi>h</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math>

Option 4 - 25 kmh-1

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Vishal Baghel | Contributor-Level 10

8 months ago

Correct Option - 3

Detailed Solution:

Let $V_{R G_{r}}$ $_{_{}}$ (velocity of Rain w.r.t. ground or w.r.t to standing Girl)

${\vec{V}}_{R G_{r}} = V_{x} \hat{i} + V_{y} \hat{j}$

$t a n (270 + 45 °) = - c o t 45 ° = \frac{V_{y}}{V_{x}} = - 1$

Now, velocity of Rain w.r.t. moving Girl

${\vec{V}}_{R G_{m}} = (V_{x} - 15 \sqrt[]{2}) \hat{i} + V_{y} \hat{j} = (15 \sqrt[]{2} - 15 \sqrt[]{2}) \hat{i} + V_{y} \hat{j} = V_{y} \hat{j}$

${\vec{V}}_{R G_{m}} = - 15 \sqrt[]{2} \hat{j} \Rightarrow {\vec{V}}_{R G_{m}} = \frac{- 30}{\sqrt[]{2}} \hat{j} \Rightarrow | {\vec{V}}_{R G_{m}} | = \frac{30}{\sqrt[]{2}}$