How does aircraft weight affect landing distance? I am a private pilot. I posed a question to instructors in the flight school and got differing answers. For a given aircraft, landing at the same speed, with everything the same except for landing weight, and using maximum braking without skidding the tires, which will stop in a shorter distance after the wheels touch down:  the plane at max gross weight, or the same plane lightly loaded? 

I submit that because the heavy plane has more energy to dissipate, it would take longer to stop. Some instructors said that because there is more weight on the wheels, the heavier plane would stop in a shorter distance. I think it is a simple energy conversion physics issue.
If you discount aerodynamic effects, and assume the same landing speed, on a theoretical level mass makes no difference. The additional kinetic energy to be dissipated is exactly cancelled by the additional normal force. The kinetic energy is given by the formula:

E = 1/2 mv², where


E is the kinetic energy
m is the vehicle's mass
v is the speed at the start of braking


The work done by braking is given by:

W = μmgd, where 


W is work done
μ is the coefficient of friction between the road surface and the tires
g is the gravity of Earth
d is the distance travelled


The braking distance given an initial speed v is then found by putting W = E, from which it follows that

d = v²/2μg 


d = minimum stopping distance
v = velocity when brakes were applied
μ = coeff. of friction between wheels and road
g = gravitational constant


Mass is removed from the equation. The kinetic energy is tranferred into heat, so the larger the mass the more heat is generated. But theoretically the stopping distance is unaffected.

In reality, though, the coefficient of friction of tires does not remain constant with changing normal force. As normal force increases the coefficient of friction (μ) of tires decreases. 

Since the μ does not remain linear with mass a heavier plane will have an increase in kinetic energy which is higher than the corresponding increase in friction of the tires, thus a longer stopping distance. 

All of this assumes that “maximum braking” means you have as much brake power available as the maximum friction force of the tires. Above a certain weight/speed this will not be the case for an airliner. They are designed for a particular amount of deceleration force at MTOW and that will be less than the amount of friction tires can provide on a dry runway. (See this question for more details on that.) That will also cause a heavier plane to roll farther.