Question

A bird is attempting to fly northeast at a constant speed, but a wind blowing southward at 5 miles per

hour blows the bird off course. If the bird’s overall movement (incorporating its intended movement

and the movement due to wind) is at a speed of √53 miles per hour, how fast would it have been traveling if there was no wind?

Answer 1

Answer:

The value is $v_b = 9.89 \ miles /hour$

Step-by-step explanation:

From the question we are told that

The velocity of the wind southward is $v = 5 j \ miles / hour$

The resultant velocity of the bird with the with wind is $V= \sqrt{53}$

Generally for an object moving in the northwest direction the angle with the horizontal is 45°

Generally the velocity of the bird in the along the x -axis is

$V_x= v_b cos 45^o i$

Generally the velocity of the bird in the along the y -axis is

$V_y=(v_b sin 45^o - 5)j$

Here $v_b$ is the velocity of the bird without the wind

Generally the resultant velocity of the bird with the with wind is mathematically represented as

$V = \sqrt{V_x^2 + V_y^2 }$

=> $\sqrt{53} = \sqrt{(v_b cos 45^o)^2 + (v_b sin 45^o - 5)^2 }$

Generally

$sin 45^o = \frac{1}{\sqrt{2} }$

and

$cos 45^o = \frac{1}{\sqrt{2} }$

So

$\sqrt{53} = \sqrt{(v_b* (\frac{1}{\sqrt{2} ))^2 + ([v_b * (\frac{1}{\sqrt{2} ) ]- 5)^2 }$

=> $53 = \frac{1}{2} v_b^2 + \frac{1}{2} v_b^2 + 5^2 -2*5 * \frac{1}{\sqrt{2} } v_b$

=> $53 = v_b^2 + 25 - 5 \sqrt{2} v_b$

=> $v_b^2 - 5 \sqrt{2} v_b -28 = 0$

Solving the above quadratic equation using quadratic formula we obtain that

$v_b = 9.89 \ miles /hour$

The other value is negative so we do not make use of it because we know that the bird is moving in the positive x and y axis