Many pilots find the concept confusing, so don't feel you're alone. <g> Perhaps I can help clear it up.
First, we must understand that V speeds are based on "balanced field length". This has NOTHING to do with the amount of runway available. It simply means that the distance to accelerate-stop equals the distance to accelerate-go.
Second, if you are heavier, it will take longer to stop. But, it will also take longer to "Go". So, your balanced field length will increase.
When the balanced field length equals the runway available (some carriers/countries also figure stopways and clearways) you are at maximum t/o weight. When there is an increase in density altitude, you decrease your max t/o weight. The V (basic) speeds will change very little. They will increase approx. 2 kts. per 5000'.
It's also important to recognize that there are "basic" V speeds and "minimum" V speeds. V1 min is driven by Vmcg. Vr min is driven by Vmu and Vmca, and V2 min is driven by Vmca and Vs. In no case may any of the V speeds be less than the corresponding Vmin speeds. That is why on some very short runways, you can't go even if you're very light.
AT higher density altitudes, many of the Vmin speeds will decrease due to the lesser thrust output of the engines, therefore lowering Vmcg and Vmca.
Actually, V1 basic and V2 basic will INCREASE with higher density altitudes (either temp or pressure altitude). It is the V1 and V2 MINIMUM speeds that decrease. So, if you are limited by V1 min, for example, and you find the temperature has increased, you will see a reduction in V1. Recognize also, that your max t/o weight will be reduced.
To summarize: You determine the t/o weight first. Then get you V speeds corresponding to it. I bet you'll find that the speeds for a 10,000' runway will be the same as for a 20,000' runway.
P.S. Sure, in real life the X-wind will have an effect on Vmcg. The trick is in knowing WHICH engine will fail!
------------------
[This message has been edited by quid (edited 15 September 2000).]