To determine the resultant velocity of the plane when it encounters a crosswind, we can apply the Pythagorean theorem. The reason for this is that the velocity of the plane and the crosswind form a right triangle, where the plane's velocity and the wind speed are perpendicular.

Given:

• Speed of the plane with respect to the air: $80 \, \text{mph}$
• Speed of the crosswind: $15 \, \text{mph}$

The resultant velocity $V_r$ is the hypotenuse of this right triangle. Using the Pythagorean theorem:

$V_r = \sqrt{(80)^2 + (15)^2}$

Let’s calculate that.

$V_r = \sqrt{6400 + 225} = \sqrt{6625} \approx 81.41 \, \text{mph}$

Result:

The resultant velocity of the plane is approximately 81.41 mph.

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Would you like further clarification on any part of the calculation?

Here are 5 follow-up questions related to this topic:

1. What if the crosswind speed increases? How would it affect the resultant velocity?
2. How would a headwind or tailwind affect the plane's resultant velocity?
3. Can you calculate the direction (angle) of the resultant velocity?
4. How does air resistance factor into real-world flight conditions?
5. What are some examples of how pilots adjust for crosswinds during takeoff and landing?

Tip: In real-world scenarios, pilots often use trigonometric calculations to adjust the heading of the plane to compensate for crosswinds and maintain the desired course.