i'm at 18k. every time i drive the car from start, the car after 5-10 minutes, at the stop light a fan comes on (it is audible) then a "push" occurs or a jerk. i do not know if transmission is to blame, but what is this fan sound coming on? (btw, weather is around 40F). when i mention the car pushes forward, i mean that if my foot is not glued tight on to the brake and on it hard, that car will move forward.
The fan noise is the cooling fan for the radiator; in the ambient temps you describe, that's about right for the coolant temp to hit the point where the fan will be needed to keep the coolant at a steady temperature.
My guess is that the slight push you're feeling is probably the compressor cycling in the climate control. Are you running the climate control on Automatic? If so, the compressor will cycle and when it does, it has the effect of slightly increasing engine rpm, which will be noticeable if you are stopped with light pressure on the brake pedal.
i don't use auto climate. but i also think it could be the radiator fan. the push is annoying. it seems that if i let the car warm up for 3 minutes, the push is not there anymore. now that is a waste of gas. btw, why would the fan turn on in 40F? that is really cool temperature.
Call up your temp display on the dash; have it showing as the car warms up. No matter what the outside temp, the coolant will hit 80-90 Celsius in a matter of minutes, depending on how you're driving. The radiator fan will come on as soon as the coolant temp gets in this range. Short of driving in Artic temps, the engine needs cooling air moved through the radiator. How much and how fast depends on outside temps, but the variance is only a few minutes on way or the other.