At low altitude engine loss you have no choice but to land straight ahead. Anything but the slightest heading adjustments will most probably end catastrofically. Your memory items are (at least, depends on complexity of aircraft):
Gently but decisevily lower the nose
Close fuel valve
Adjust airspeed for landing
Flare and land
How do you determine if you have enough runway ahead? That's a tricky one, and in this situation you quite possibly have milliseconds to make that determination. It's going to be an estimate at best because you certainly wont have time to do any calculations. As a general rule I'd say if you have around half the runway length left you're probably good.
With an emergency, the objective is not to fly the quickest pattern possible, it's to be back safely on the ground as soon as possible, with the greatest probability of success. If you save 10 seconds by doing something that substantially increases the risk of wrecking the aircraft because you flew it into something, that's probably a poor tradeoff.
In the ...
The quickest and safest way is to fly the standard pattern, especially if you are at an uncontrolled airport. Why? Because that is where everyone else is looking for you to be. Per the Airplane Flying Handbook Chapter 7 states:
compliance with the basic rectangular traffic pattern reduces the
possibility of conflicts at airports without an operating ...
Temperature and humidity both affect air density and engine power. $H_2O$ molecules are lighter than the $N_2$ and $O_2$ molecules they displace in a volume of humid air.
1. Air Density. The graph below from this site shows that at 40°C, which is a high atmospheric temperature, dry air is about 0.94 density at 20°C, while 100% humidity reduces the density to ...