I'm struggling to figure out why that is the case. I'm trying to determine why the best angle of climb speed equals to best rate of climb speed at the absolute ceiling on airplanes.
$\begingroup$ Can you elaborate? What are you talking about? Vertical speed? Airspeed? What aircraft? Also please put your question in the body of your post. $\endgroup$– Ron BeyerJan 14, 2017 at 4:54
1$\begingroup$ Vx as in the best angle of climb speed and Vy as in best rate of climb speed, so airspeed. Airliners was the aircraft I was directing this towards. Sorry. $\endgroup$– nyorkr23Jan 14, 2017 at 4:58
$\begingroup$ Presumably indicated speeds for best angle and best rate of climb. $\endgroup$– DJClayworthJan 14, 2017 at 4:59
I think the answer to your question lies in the fact that approaching the absolute ceiling, the terms "climb rate" and "angle of climb", in regards to an airspeed, begin to lose their meanings.
As a plane approaches the absolute ceiling, the climb rate will be zero or close to zero, so the best angle of climb approaches 0 degrees above the horizontal plane as well making your "best-rate" speed also be your "best angle" speed; both the rate of climb and thus the angle of climb being, effectively, zero.
Any speed which yields less than the best rate of climb (if you want to call it that) of zero at the absolute ceiling would be causing a descent, and thus a negative angle of climb as well making any speed other than the best rate of climb speed also not be the best angle of climb speed, which is why that is the altitude at which best-rate and best-angle speeds coalesce.
$\begingroup$ I agree with the answer in that if you've reached your ceiling then you're not going up; Vx, Vy. $\endgroup$ Jan 14, 2017 at 18:23
$\begingroup$ Yeah, exactly. Your best rate and angle of climb basically become, "not descending", and you just so happen to be going a certain speed therefore, they've coalesced. This may be a bit of a sensationalist answer for some people with advanced knowledge like design engineers, but I think for the average Joe, this way of explaining it will help them gain an understanding and "wrap their mind around" the concept. $\endgroup$ Apr 14, 2018 at 2:34
The absolute ceiling is defined as the altitude where climb speed drops to zero. This implies that there is only one speed at which the airplane does not lose altitude. Flying any faster or slower will result in higher drag than thrust, and the airplane will start to sink. Only at the optimum climb speed will the airplane stay at zero sink speed. Now that there is only one speed at which both the climb angle (which is now zero) and the climb speed (which is also zero) reach their maximum, both speeds must coalesce.
Climb optima for different thrust loadings (own work). Flight speed is plotted on the X axis while climb speed is plotted on the left Y axis. Green lines show sink speed while blue lines show thrust (right Y axis).
This graph shows the idealized optima of best climb speed and best climb angle (steepest climb) for different thrust loadings of a turbofan-powered aircraft (full explanation here). Note that the lines for the two optima cross at zero climb speed. It is no coincidence that the sink speed curve over flight speed for zero thrust loading looks like the glide polar of a glider.