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Question: A helicopter is to reach a point 200000m east of his existing place. Its velocity relative to wind b...

A helicopter is to reach a point 200000m east of his existing place. Its velocity relative to wind blowing at 30kmh−130km{h^{ - 1}} from northwest taking schedule arrival time duration as 40 minute is
A. 2i^+279j^2\hat i + 279\hat j
B. 279i^+21j^279\hat i + 21\hat j
C. 729i^+12j^729\hat i + 12\hat j
D. 12i^+729j^12\hat i + 729\hat j

Explanation

Solution

Here, in this type of question we will use the concept of relative velocity. We have given the velocity of wind with respect to earth and now according to given distance we will find out the velocity of the helicopter with respect to earth. And finally we will calculate the resultant velocity of the helicopter with respect to wind.

Complete step by step answer:
Let v⃗HE{\vec v_{HE}} be the velocity of the helicopter with respect to earth.
∴\therefore we all know that velocity is equal to displacement upon time.
Mathematically- velocity = displacementtime{\text{velocity = }}\dfrac{{displacement}}{{time}}
So given displacement is 200000m and time is 40 minutes
So,
v⃗HE=2000001000×4060{\vec v_{HE}} = \dfrac{{200000}}{{1000 \times \dfrac{{40}}{{60}}}} displacement is converted into kilometer and time is converted into hours
v⃗HE=20000010006040 v⃗HE=300 kmh−1  {{\vec v}_{HE}} = \dfrac{{200000}}{{1000}}\dfrac{{60}}{{40}} \\\ {{\vec v}_{HE}} = 300{\text{ }}km{h^{ - 1}} \\\
Vector representation of helicopter velocity will be-
v⃗HE=300i^{\vec v_{HE}} = 300\hat i
Now we have given velocity of wind with respect to earth is 30 kmh−130{\text{ km}}{{\text{h}}^{ - 1}}from northwest i.e. the wind is blowing in south-east direction
Let v⃗WE{\vec v_{WE}} be the velocity of earth with respect to earth.
So, vector representation of velocity of wind with respect to earth is given by;

This above written velocity is the component of wind velocity in horizontal and vertical direction
Now we have to find the resultant velocity.
Let v⃗HW{\vec v_{HW}} is velocity of helicopter with respect to wind
So, v⃗HW=v⃗HE−v⃗WE{\vec v_{HW}} = {\vec v_{HE}} - {\vec v_{WE}} where meaning of all symbol is as discussed before
On putting value of variables:
So,
On solving above equation:
v⃗HW=300i⃗−(152i^−152j^){\vec v_{HW}} = 300\vec i - (15\sqrt 2 \hat i - 15\sqrt 2 \hat j)
On applying resultant formula for subtraction
v⃗HW=279i^+21j^{\vec v_{HW}} = 279\hat i + 21\hat j
So, correct option will be B

Note: Here we have applied the concept of relative velocity. Relative velocity is the velocity of an object with respect to another. In relative velocity concept we use the concept of frame of reference. We use inertial frame of reference and non-inertial frame of reference. When we talk about velocity with respect to earth this is called the inertial frame of reference and when we talk about velocity with respect to other objects it is called non-inertial frame of reference.