A plane is flying due north at 160 km/h relative to the surrounding air. There is a crosswind blowing due east. If the magnitude of the resultant velocity of the plane is 200 km/h,... A plane is flying due north at 160 km/h relative to the surrounding air. There is a crosswind blowing due east. If the magnitude of the resultant velocity of the plane is 200 km/h, what is the speed of the crosswind?
Understand the Problem
The question involves determining the speed of a crosswind affecting a plane's flight path. The plane is flying north, and a wind is blowing east, creating a resultant velocity. We need to apply the Pythagorean theorem to find the speed of the crosswind based on the given velocities.
Answer
The speed of the crosswind is $120 \, \text{km/h}$.
Answer for screen readers
The speed of the crosswind is $120 , \text{km/h}$.
Steps to Solve
- Identify known and unknown quantities
We have the following velocities:
- Velocity of the plane flying north, $V_p = 160 , \text{km/h}$.
- Magnitude of the resultant velocity, $V_r = 200 , \text{km/h}$.
- Velocity of the crosswind, $V_w$, is unknown.
- Apply the Pythagorean theorem
The plane's velocity and the crosswind velocity are perpendicular to each other. Hence, we can use the Pythagorean theorem:
$$ V_r^2 = V_p^2 + V_w^2 $$
Substituting the known values:
$$ (200 , \text{km/h})^2 = (160 , \text{km/h})^2 + V_w^2 $$
- Calculate the squares
Calculate the squares of the velocities:
$$ 40000 , \text{km}^2/\text{h}^2 = 25600 , \text{km}^2/\text{h}^2 + V_w^2 $$
- Rearrange the equation
Now, rearranging the equation to isolate $V_w^2$:
$$ V_w^2 = 40000 , \text{km}^2/\text{h}^2 - 25600 , \text{km}^2/\text{h}^2 $$
- Perform the subtraction
Calculate the right side:
$$ V_w^2 = 14400 , \text{km}^2/\text{h}^2 $$
- Take the square root
Now take the square root to find $V_w$:
$$ V_w = \sqrt{14400} , \text{km/h} $$
- Calculate the speed of the crosswind
The final step is to compute the square root:
$$ V_w = 120 , \text{km/h} $$
The speed of the crosswind is $120 , \text{km/h}$.
More Information
The calculation demonstrates how to apply the Pythagorean theorem in a real-world situation involving vector velocities. The crosswind affects the plane's actual path and can be critical for navigation.
Tips
- Mixing up the velocities; ensure that the correct values are squared and added according to the Pythagorean theorem.
- Forgetting to take the square root when finally extracting the crosswind speed.
AI-generated content may contain errors. Please verify critical information
