The aircraft at high altitude will fly more slowly than at sea level, but this is not simply a function of power available. In order to understand the way in which increasing altitude affects airspeed we need to consider the following factors:
RELATIONSHIP BETWEEN INDICATED AIRSPEED AND DRAG
The airspeed indicator gives an output (IAS) that is proportional to dynamic pressure (1/2 Rho V squared), where Rho is air density and V is the true airspeed (TAS). If we ignore the complexities of compressibility, then any given value of (1/2 Rho V squared) will always give the same IAS at all altitudes. Airspeed indicators are calibrated to give IAS equal to TAS at sea level in the International Standard Atmosphere.
Drag is also proportional to (1/2 Rho V squared) so the drag force will be constant at any given IAS at all altitudes. So if our aircraft climbs at constant IAS it means that it climbs at constant (1/2 Rho V squared) and constant drag.
RELATIONSHIP BETWEEN INDICATED AIRSPEED AND TRUE AIRSPEED
As altitude increases, air density decreases. But climbing at constant IAS means climbing at constant (1/2 Rho V squared). This means that in order to maintain constant IAS and (1/2 Rho V squared), the value of V squared must increase to balance the rate of decrease in Rho.
At an altitude of 40000 feet in the International Standard Atmosphere, Rho is approximately ¼ of its sea level value. So if an aircraft climbs from sea level to 40000 feet, the decrease in Rho to ¼ of its starting value must be matched by an increase in V squared to 4 times its starting value. But V squared is TAS squared. If TAS squared increases by a factor of 4, then TAS must increase by a factor of 2. This means that at 40000 feet the TAS is approximately twice the IAS.
RELATIONSHIP BETWEEN ALTITUDE AND POWER REQUIRED
Power required is equal to drag multiplied by TAS. So when climbing at any given IAS, although the drag remains constant, the power required increases due to the increasing TAS. This means that at 40000 feet in the International Standard Atmosphere, the power required is greater than at the same IAS at sea level.
To find out how much the power required has increased we need to look again at the drag equation. Drag is proportional to (1/2 Rho V squared) where the V is TAS. When we multiply drag by TAS to get power required, we have TAS squared multiplied by TAS, which is TAS cubed. This means that power required is proportional to TAS cubed.
But at 40000 feet the TAS at any given IAS is twice its sea level value. The power required at 40000 feet must therefore be 2 cubed (or 8) times its sea level value. So flying at any given IAS requires 8 times as much power at 40000 feet as it does at sea level.
RELATIONSHIP BETWEEN ALTITUDE AND POWER AVAILABLE
The power output of any aircraft propulsion system is proportional to the mass flow of air passing through it. If we (incorrectly) assume that our engines are running at maximum RPM at sea level, then as we climb reducing air density will reduce both air mass flow and power output. At 40000 feet where the air is on ¼ of its sea level density we would have approximately ¼ of our maximum sea level power available. In reality the RPM would increase with increasing altitude. This would partly offset the reduction in air density, but would eventually become impracticable when the limiting RPM or EGT were reached. The overall effect would therefore be a reduction in power available as altitude increased.
BRINGING THESE FACTORS TOGETHER
The TAS and power required at any given IAS are much greater at 40000 feet than at sea level.
The power available is much less at 40000 feet.
Even if power available were to remain constant, the increasing power required would cause both TAS and IAS to decrease with increasing altitude.
The increase in power required is more significant than the decrease in power available, in causing airspeed to decrease as altitude increases.