Question

To a man walking at a rate of 4kmph the rain appears to fall vertically. when he doubles his speed the rain appears to meet him at an angle of 45° with the vertical. find the speed of rain.​

Answer 1

Given:

Initial speed of the man = 4 km/h

Final speed of the man = 8 km/h

To find:

The speed of the rain

Calculation:

Let us assume i and j as unit vectors along horizontal and vertical directions respectively

Let the velocity of the rain   $V_r=a\hat{i}+b\hat{j}$

Initially the man is walking with a speed of 4 km/h and the rain appears to fall vertically

  $V_{rm}=V_r-V_m$

  $V_{rm}=(a\hat{i}+b\hat{j})-4\hat{i}$

  $V_{rm}=(a-4)\hat{i}+b\hat{j}$

As relative velocity is vertical

  $a-4=0$  

  $a=4$

After the man is walking with a speed of 4 km/h and the rain appears to fall at an angle of 45°

  $V_{rm}=V_r-V_m$

  $V_{rm}=(4\hat{i}+b\hat{j})-8\hat{i}$

  $V_{rm}=(-4)\hat{i}+b\hat{j}$

We know that

  $Tan\theta=\frac{b}{-4}$

  $Tan(45^{\circ})=-\frac{b}{4}$

  $b=-4$

Let the velocity of the rain is given by $V_r=4\hat{i}+-4\hat{j}$

Speed of the rain  $=\sqrt{4^{2}+4^{2}}$

                              $=\sqrt{32} =4\sqrt{2}\ m/s$

Final answer:

The speed of the rain is 4√2 m/sec

Answer 2

Answer:

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