Name: Motion in a Plane With Constant Acceleration
Uploaded: 2019-07-19
Duration: 6 min 42 s
Description: In this video, we are going to look at the simple concept of motion in a plane with constant acceleration. You'll be able to master the concept in just a few...

Question

A helicopter flies horizontally with constant velocity in a direction $θ$ east of north between two points $A$ and $B$ , at distance $d$ apart. Wind is blowing from south with constant speed $u$ , the speed of helicopter relative to air is $n u$ , where $n > 1$ . Find the speed of the helicopter along $A B$ . The helicopter returns from $B$ to $A$ with same speed $n u$ relative to air in same wind. Find the total time for the journeys.

Accepted Answer

Given,

$A$

Velocity of helicopter w.r.t. wind is given by,

${\vec{v}}_{HA} = {\vec{v}}_{H} - {\vec{v}}_{A}$

$\Rightarrow |{\vec{v}}_{HA}| = n u = \sqrt{u^{2} + v^{2} - 2 u v \cos (θ)}$

$\Rightarrow n^{2} u^{2} = u^{2} + v^{2} - 2 u v cosθ$
$\Rightarrow v^{2} - 2 u v cosθ = u^{2} (n^{2} - 1)$
$u$

$\Rightarrow v = u cosθ \pm u \sqrt{n^{2} - \sin^{2} θ}$

As $n \geq 1$ ,

$n^{2} - \sin^{2} θ \geq 1 - \sin^{2} θ \geq \cos^{2} θ$

Hence, speed of the helicopter is $v = u cosθ + u \sqrt{n^{2} - \sin^{2} θ}$ .

Now we consider motion from $B$ to $A$ .

Similar to first case, the helicopter will return to $A$ along $B A$ when its velocity after being deflected by wind is along $B A$ . Once again using the cosine rule in the vector triangle, we get

$Let |{\vec{V}}_{H}| = v_{2}$

${\vec{v}}_{HA} = {\vec{v}}_{H} - {\vec{v}}_{A}$

$\Rightarrow |{\vec{v}}_{HA}| = n u = \sqrt{u^{2} + v_{2}^{2} - 2 u v_{2} \cos (180 ° - θ)}$

$\Rightarrow n^{2} u^{2} = u^{2} + {v_{2}}^{2} + 2 u v_{2} cosθ$

$\Rightarrow v_{2} = - u cosθ + u \sqrt{n^{2} - \sin^{2} θ}$

The total time for motion $A$ to $B$ and $B$ to $A$ ,

$t = \frac{d}{v} + \frac{d}{v_{2}} = \frac{d (u + v_{2})}{u v_{2}} = \frac{2 d u \sqrt{n^{2} - \sin^{2} θ}}{u^{2} (n^{2} - \sin^{2} θ) - u^{2} \cos^{2} θ}$
$\Rightarrow t = \frac{2 d \sqrt{n^{2} - \sin^{2} θ}}{u (n^{2} - 1)}$

Important Questions on Kinematics II

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