an airplane flying due north with a speed relative to the ground of 300 km/hr. it then encounters a cross-wind that is directed due east. of the resultant speed of the airplane is now 302 km/hr, how fast is the cross wind?
yes Pythagorean theorem
302^2=300^2 + V^2
To solve this problem, we can use the concept of vectors and the Pythagorean theorem.
Let's assign values to the given information:
The speed of the airplane relative to the ground (in the north direction) = 300 km/hr.
The resultant speed of the airplane after encountering the crosswind = 302 km/hr.
Now, let's form a right-angled triangle to represent this situation, with the velocity due north as the vertical component and the crosswind velocity as the horizontal component.
Let's denote:
The speed of the crosswind = V_crosswind
The speed of the airplane relative to the crosswind = V_airplane
Using the Pythagorean theorem, we have:
(Resultant speed of the airplane)^2 = (Speed of the crosswind)^2 + (Speed of the airplane relative to the crosswind)^2
(302 km/hr)^2 = (V_crosswind)^2 + (V_airplane)^2
Simplifying this equation, we have:
(302 km/hr)^2 - (300 km/hr)^2 = (V_crosswind)^2
91,204 km^2/hr^2 - 90,000 km^2/hr^2 = (V_crosswind)^2
1,204 km^2/hr^2 = (V_crosswind)^2
Now, to find the speed of the crosswind, we take the square root of both sides:
V_crosswind = √(1,204 km^2/hr^2)
V_crosswind ≈ 34.7 km/hr
Therefore, the speed of the crosswind is approximately 34.7 km/hr.
To determine the speed of the crosswind, we can use vector addition.
Let's assume the speed of the airplane (relative to the air) is A km/hr, and the speed of the crosswind is C km/hr.
The airplane's speed relative to the ground can be obtained by considering the horizontal and vertical components of its motion separately.
Since the airplane is flying due north with a speed of 300 km/hr relative to the ground, the vertical component of its motion, which is in the north direction, is also 300 km/hr.
The horizontal component of its motion is caused by the crosswind. Therefore, the speed of the airplane in the eastward direction due to the crosswind is C km/hr.
Now, the total speed of the airplane relative to the ground is given as 302 km/hr.
Using the Pythagorean theorem, we can express the magnitude of the total speed as the square root of the sum of the squares of its horizontal and vertical components:
(300 km/hr)^2 + (C km/hr)^2 = (302 km/hr)^2
Simplifying the equation:
90000 + C^2 = 91204
C^2 = 91204 - 90000
C^2 = 1204
Taking the square root of both sides yields:
C ≈ 34.7 km/hr
Therefore, the speed of the crosswind is approximately 34.7 km/hr.