# Help in understanding this velocity triangle of a propeller

I was reading a research paper and have had difficulty understanding something. How did the author get tan(phi) in these terms?

This is something I've been trying to figure out for a long time now.

• This is a question about Trigonometry, not Aviation, – Ralph J Dec 3 '20 at 15:16
• Yeah exactly. But by basic trigonometry this relationship can't be obtained. I figured it has to do something with velocity triangle of propeller wing so posted here! – Shivam Sundriyal Dec 3 '20 at 21:05
• V_propeller, V2, and tan(phi) are related as a basic trig problem. What else are you looking for? – Ralph J Dec 3 '20 at 21:37

There is an obvious error in this paper. Just look at the final formula, and remember that for small angles $$\tan(\phi) \sim \phi$$. The formula says that if you increase $$\phi$$, $$v_2$$ decreases. Of course, the opposite happens.

By (trigonometric) definition,

$$\tan(\phi) = \frac{v_2}{v_{propeller}}$$

Therefore,

$$v_2 = v_{propeller} \cdot \tan(\phi)$$

The rest follows; the result is similar but you have multiplication instead of the fraction.

P.S. You can get a hint that this is a low quality paper by the fact that they obliviously confuse upper- and lowercase $$v$$ (on the picture and in the formulae). In math/science, case matters! Sometimes even the script matters.

• Thanks for the answer. I got this from the paper DESIGN AND DEVELOPMENT OF A HYBRID DRONE by British Columbia Institute of technology. They had various professors with PhD's working on it so I thought they wouldn't have made such a basic mistake! Can you look up their report once? – Shivam Sundriyal Dec 4 '20 at 10:08