Angular and linear velocity propeller

[henry3369]

Homework Statement

You are designing an airplane propeller that is to turn at 2400 rpm (Fig. 9.13a). The forward airspeed of the plane is to be 75m/s, and the speed of the tips of the propeller blades through the air must not exceed 270m/s. (This is about 80% of the speed of sound in air. If the speed of the propeller tips were greater than this, they would produce a lot of noise.) What is the maximum possible propeller radius?

Homework Equations

v_tip² = v_plane² + v_tan²

The Attempt at a Solution

I know how to solve this equation to find the maximum radius, but I'm having trouble understanding the derivation of the equation. I know that v_plane points in the direction of the plane's motion; thus, the propeller also has this same velocity in the same direction. Then angular velocity can be related to tangential velocity by v = rw. How does the vector sum of these two result in the velocity of the tip of the propeller and isn't the direction of v_tip also be tangential to the direction of motion of the propeller so v_tip = v_tan?

 

[BvU]

Tip trajectory is a spiral in the air. The 75 m/s does contribute (a little) in the square root of the squared sum.