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

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

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= √(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 = √(V_x^2 + V_y^2 )`

=>
`√(53) = √((v_b cos 45^o)^2 + (v_b sin 45^o - 5)^2 )`

Generally

`sin 45^o = (1)/(√(2) )`

and

`cos 45^o = (1)/(√(2) )`

So

=>
`53 = (1)/(2) v_b^2 + (1)/(2) v_b^2 + 5^2 -2*5 * (1)/(√(2) ) v_b`

=>
`53 = v_b^2 + 25 - 5 √(2) v_b`

=>
`v_b^2 - 5 √(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