Why does "V1 increase with decreasing density"?

Question

The statement made in the question is mentioned in one of the renowned publications.

My contradictions which I have already ruled out as a potential solution:

• As Density decreases Vmcg decreases thus V1 is not limited by Vmcg.

• As Density decreases Vmbe decreases as there's insufficient cooling of the brakes, thus V1 should ideally be limited by Vmbe.

• At lower density the aircraft needs to be at a higher groundspeed to achieve the same CAS, but the ASDA would be limiting factor as it would require a greater stopping distance, this would again limit V1.

Answer 1

I guess the statement is about V1 from the balanced field length, which is where the accelerate-go distance is equal to the accelerate-stop distance:

^{(image source: boldmethod > Engine Failure On Takeoff: Do You Stop Or Go?)}

How do these curves change when the density decreases? Lower density means less thrust, therefore accelerating will be slower and uses more distance. This affects both curves equally for the acceleration to V1, but afterwards only the accelerate-go distance:

• Let us call the force accelerating the aircraft initially $ F = 2 \times T $ (thrust from two engines).
• In the accelerate-go case, you would only accelerate with a force of $ F = T $ after the engine failure.
• In the accelerate-stop case, you brake with a certain force $ F = F_\text{brake} $.
• Now, at lower density, the thrust $ T $ will decrease, but the brake force $ F_\text{brake} $ will be unaffected.
• The accelerate-go case is therefore affected during the entire time by the lower value of $ T $.
• The accelerate-stop case is only affected during the accelerate part, but afterwards the brake force is still the same.

Therefore, both curves will move up in the diagram, but the accelerate-go distance will move up more. This implies that their intersection (the balanced field length) will occur further to the right, resulting in a higher value for V1.

Answer 2

We all know that V1 is limited by both the Vmcg (minimum control speed-ground) and the Vmbe (maximum brake energy). It is also affected by Vr (rotation speed). But that is for some other question. Firstly, V1 cannot be lower than Vmcg because if an engine fails below V1, the air flow over the vertical stabilizer and the rudder is not sufficient enough to control the aircraft. On the other hand, V1 cannot be higher than Vmbe because, if you try to stop the aircraft at a higher speed than Vmbe, the brakes can fail and in worst case catch fire.

If the density is lower, the Vmcg reduces because less thrust is produced by the engines. So, in an engine failure the amount of force that needs to be counteracted by the rudder against the live engine reduces. As, you have mentioned thus with reduced densities, Vmcg does not limit the V1.

A reduction in density reduces the value of Vmbe as well. So, if we consider a balanced field, the value of V1 will be higher than Vmbe. As we have discussed before, this is not an ideal situation. To ensure a safe take off hence, the value of V1 is reduced until it becomes less or equal to Vmbe. This will increase the TODR (Take off distance required) and reduce the ASDR (Accelerate stop distance required). Why does this happen? Any time, the V1 is reduced, the ASDA decreases and the TODR increases. The ASDA decreases because the lower the V1, the earlier it is assumed an engine will fail, so the less the aircraft is accelerated. Therefore, the lesser the acceleration, the shorter the aircraft can be stopped. The TODR increases because a reduced V1 just like before results in a reduced acceleration. It simply takes more distance to take off with one engine operative than with two.

To answer your question. At a lower density, the reduction in Vmbe means, you will have to reduce the value of the V1 to that of Vmbe to get the best field performance. The result is an increased TODR and a decreased ASDA. The total field length required increases because it is limited by the TODR.

The dark dotted line is the ideal V1 which occurs at the balanced field (ASDA = TODR). Here, the value of Vmbe is reduced due to a decrease in density. So, if you reduce the V1 to that of Vmbe, the TODR increases and the ASDA decreases. The yellow line is the total field length and it increases to account for the TODR.