An airplane pilot wishes to fly due west. A wind of 80.0 km/h (about 50 mi/h) is blowing toward the south. What is the speed of the plane over the ground? Draw a vector diagram.

Verified step by step guidance

Begin by identifying the vectors involved: the airplane's velocity vector and the wind's velocity vector. The airplane's velocity vector points due west, while the wind's velocity vector points south.

Draw a vector diagram. Represent the airplane's velocity vector as a horizontal arrow pointing to the left (west) and the wind's velocity vector as a vertical arrow pointing downward (south). The resultant vector, which represents the plane's velocity over the ground, will be the diagonal of the rectangle formed by these two vectors.

Use the Pythagorean theorem to find the magnitude of the resultant vector. If the airplane's velocity is 'v' km/h and the wind's velocity is 80.0 km/h, the magnitude of the resultant vector (ground speed) can be calculated using the formula:

\sqrt{v^{2} + {80.0}^{2}}

Determine the direction of the resultant vector using trigonometry. The angle θ between the resultant vector and the westward direction can be found using the tangent function:

\tan θ = \frac{80.0}{v}

. This will give you the angle south of west.

Summarize the findings: The speed of the plane over the ground is the magnitude of the resultant vector, and its direction is given by the angle calculated. This completes the vector analysis of the airplane's motion considering the wind.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Addition

Vector addition is a method used to combine two or more vectors to determine a resultant vector. In this context, the airplane's velocity vector and the wind's velocity vector must be added to find the plane's actual velocity over the ground. This involves using the Pythagorean theorem or trigonometric functions to resolve the vectors into components and calculate the resultant vector.

Relative Velocity

Relative velocity is the velocity of an object as observed from a particular reference frame. Here, the plane's velocity relative to the ground is affected by the wind's velocity. To find the ground speed, we must consider the plane's intended velocity and adjust for the wind's influence, effectively treating the wind as a moving reference frame.

Trigonometry in Physics

Trigonometry is essential in physics for resolving vectors into components and calculating angles and magnitudes. In this problem, trigonometric functions like sine, cosine, and tangent help determine the resultant vector's magnitude and direction. By constructing a right triangle with the vectors, we can use these functions to solve for the plane's speed over the ground.