An airplane flies due north at 145 km/h with respect to the air. There is a wind blowing at 65 km/h to the east relative to the ground. What is the plane's speed relative to the ground?
10 west
To find the plane's speed relative to the ground, we need to consider the horizontal and vertical components separately.
Given:
- Airplane's speed (relative to the air) = 145 km/h
- Wind speed (relative to the ground) = 65 km/h
First, let's find the horizontal component of the airplane's velocity relative to the ground.
Horizontal component: The airplane is flying due north, so there is no horizontal component due to its motion. However, we need to consider the effect of the wind on the horizontal component.
The wind is blowing eastward, while the airplane is moving northward. These two velocities are perpendicular to each other, forming a right triangle. Since we are only interested in the horizontal component, we can use the Pythagorean theorem to find it.
Using the Pythagorean theorem:
horizontal component = √(airplane's speed^2 - wind speed^2)
horizontal component = √(145^2 - 65^2) km/h
= √(21025 - 4225) km/h
= √16800 km/h
= 129.90 km/h (rounded to two decimal places)
Next, let's find the vertical component of the airplane's velocity relative to the ground.
Vertical component: The airplane is flying due north, so the vertical component is equal to its speed relative to the air.
Therefore, the vertical component of the airplane's velocity relative to the ground is 145 km/h.
Finally, we can find the plane's speed relative to the ground by combining the horizontal and vertical components.
Speed relative to the ground = √(horizontal component^2 + vertical component^2)
Speed relative to the ground = √(129.90^2 + 145^2) km/h
= √(16856.01 + 21025) km/h
= √37881.01 km/h
= 194.64 km/h (rounded to two decimal places)
Therefore, the plane's speed relative to the ground is approximately 194.64 km/h.
To find the plane's speed relative to the ground, we can use vector addition.
First, let's break down the velocities into their components. The plane's velocity due north can be represented as (0, 145) km/h (where the first component represents the east-west direction, and the second component represents the north-south direction). With the wind blowing to the east, its velocity can be represented as (65, 0) km/h.
To find the plane's resultant velocity relative to the ground, we add the components of the plane's velocity and wind velocity. Therefore, the resultant velocity can be calculated as (0 + 65, 145 + 0) km/h, which simplifies to (65, 145) km/h.
To find the magnitude of the resultant velocity (the plane's speed relative to the ground), we can use the Pythagorean theorem. The magnitude can be calculated as the square root of the sum of the squares of the components.
In this case, the magnitude of the resultant velocity is √(65^2 + 145^2) km/h. Evaluating this equation gives us √(4225 + 21025) km/h ≈ √25250 km/h ≈ 158.8 km/h.
Therefore, the plane's speed relative to the ground is approximately 158.8 km/h.