An airplane flies due north at 260 km/h with respect to the air. There is a wind blowing at 60 km/h to the northeast with respect to the ground. What are the plane's speed and direction with respect to the ground?
To find the plane's speed and direction with respect to the ground, we need to use vector addition.
Step 1: Draw a diagram
Draw a diagram representing the plane's motion. Label the north direction as "N" and the northeast direction as "NE."
Step 2: Break down velocities into components
The airplane's velocity (Vairplane) has a magnitude of 260 km/h in the north direction (N). The wind's velocity (Vwind) has a magnitude of 60 km/h in the northeast direction (NE).
Step 3: Add the components
To find the plane's velocity with respect to the ground, we need to add the two vectors together. Since the vectors are not in the same direction, we need to break them down into their x and y components.
Component of airplane's velocity (VairplaneN): 260 km/h in the north direction (N)
Component of wind's velocity (VwindNE): 60 km/h in the northeast direction (NE)
For VairplaneN, since it is in the north direction, its x-component is 0 km/h and its y-component is 260 km/h.
For VwindNE, we can find its x and y components using trigonometry.
The component in the x-direction (VwindNE_x) can be found by multiplying the magnitude of VwindNE (60 km/h) by the cosine of the angle it makes with the x-axis (45 degrees since it is northeast).
VwindNE_x = 60 km/h * cos(45°) ≈ 42.42 km/h
The component in the y-direction (VwindNE_y) can be found by multiplying the magnitude of VwindNE (60 km/h) by the sine of the angle it makes with the x-axis (45 degrees since it is northeast).
VwindNE_y = 60 km/h * sin(45°) ≈ 42.42 km/h
Step 4: Add the x and y components
To obtain the resultant x and y components, add the corresponding components from VairplaneN and VwindNE.
x-component of resultant velocity (Vx): 0 km/h + 42.42 km/h ≈ 42.42 km/h
y-component of resultant velocity (Vy): 260 km/h + 42.42 km/h ≈ 302.42 km/h
Step 5: Calculate the magnitude and direction of the resultant velocity
The magnitude of the resultant velocity (Vresultant) can be found using the Pythagorean theorem.
Vresultant = sqrt(Vx² + Vy²)
Vresultant = sqrt((42.42 km/h)² + (302.42 km/h)²)
Vresultant ≈ 306.03 km/h
The direction of the resultant velocity can be found using the inverse tangent (arctan) function.
Direction of Vresultant = arctan(Vy / Vx)
Direction of Vresultant = arctan(302.42 km/h / 42.42 km/h)
Direction of Vresultant ≈ 80.49°
Therefore, the plane's speed with respect to the ground is approximately 306.03 km/h, and its direction with respect to the ground is approximately 80.49°.