Al Bowers gave us an excellent explanation of the stability issues of a tractor vs. pusher installation. However, there are some other issues that need to be considered.
From : Don Stackhouse
What really matters in the stability issues is how far the prop is longitudunally from the C/G. For example, if you mounted the motor in the nose, with the prop running through a cutout in the wing so that the prop was located approximately at the C/G (such as that little electric B-2 model from Wattage that was popular a couple years ago), the prop would have relatively little effect on stability, despite the fact that technically speaking it was a pusher. In fact, if this "pusher" prop was ahead of the C/G, it would be just as destabilizing as a tractor prop in that location. Likewise, if you mounted the motor on the tail, such as on the "Sea Wind" amphibian, with a tractor prop on the front of it, the prop could add some stabilizing effects in pitch and yaw despite technically being a tractor installation.
Note also that most airplanes will have to fly at a range of power settings, including idle, so it's usually not wise to use the stabilizing effects of a pusher prop to reduce the size of stabilizing flying surfaces. OTOH, at least you don't have to make those surfaces bigger to counteract the destabilizing effects of a tractor prop at high power settings. Of course, if you keep a tractor prop as close as possible to the C/G , the destabilizing effects of the tractor prop tend to be fairly minor anyway. Because of all of this, the net importance of prop influenced stability issues tends to be fairly small for most aircraft, with the possible exception of flying wings.
C/G issues can be a problem. When you mount the motor in the tail, it typically takes a bunch more weight in the nose to balance it. Even if that can be arranged successfully without resorting to lots of nose ballast, you still end up with a plane that has all its mass distributed in the extremities, instead of clustered as closely as possible at the C/G. This tends to degrade the dynamic stability and the control response. There is the alternative of mounting the motor at the C/G and driving the prop through a long driveshaft, but that opens all sorts of cans of worms in terms of torsional vibrations and resonant frequencies, possible whirl-mode instability (that's where that long driveshaft decides to pretend it's a jump rope), and of course all the added weight of the drive shaft and its supporting structure.
In Al's description of the stability issues, he uses the analogy of the "box" that accelerates and deflects air, and the resulting effect on the airframe. This is fine when looking at the effects of the prop on the airframe, but you must be very careful not to let yourself fall into the trap of ignoring the effects of the airframe on the prop. There is a long and sad history of airframe designs that fell short of expectations because of exactly this kind of thinking.
A prop is more than a "box that deflects air". It is a set of rotating wings that fly through a very complex, helical, non-symmetrical flow field. Their efficiency depends a great deal on just how non axisymmetrical that flowfield is. A pusher prop has to fly through all the disturbed flow coming off of the airframe in front of it, and as a result tends to have significantly less efficiency (usually at least a couple percent less, typically quite a bit worse than that, and in some cases more than 15% less efficient) than an equivalent tractor installation. Yes, there can be some gains in airframe efficiency from not immersing as much of the airframe in the higher speed flow behind the prop, but it is usually a far smaller gain than the losses due to the airframe's detrimental effects on the prop. Only a small portion of the total airframe drag tends to be influenced by the flow of the prop, while all of the thrust from the prop can be influenced by the flow coming off the airframe. Attempts to help the airframe by putting the prop in a bad situation almost inevitably end up being "penny wise and pound foolish."
Immersing the prop in that disturbed, non axisymmetric flowfield tends to also generate much higher vibrational stresses in both the prop and the airframe. Although the lifespans of models are not usually long enough for the resulting fatigue stresses to be an issue, it can be a major problem on full scale aircraft. On models it can result in lots of weird little problems with things like screws vibrating loose all the time, wear in control linkages, etc.
Inflicting a lot of disturbed flow on the prop also tends to worsen the noise, both inside and outside the aircraft. I have some tape recordings of the Lear Fan, a turbine powered pusher, taking off. Despite having the engine exhausts about 6 feet ahead of the prop, it sounded just like the RR Merlin in a P-51 Mustang. Pushers with the exhausts closer to the prop tend to sound more like a chain saw.
Prop efficiency is also very dependent on diameter, which in turn tends to be set by ground clearance issues in many cases. The ground clearance at rotation on takeoff on a pusher tends to be the limiting factor in many cases, and the resulting diameter tends to be less than what the same airplane can handle in a tractor prop. This often results in further losses of efficiency.
In the case of a small model that gets hand-launched, the ground clearance problems are less significant. However, you do have to worry about getting your hand sliced and diced at the moment of release.
Pylon-mounted props, such as the Lake Amphibian, tend to have fuselage clearance issues that limit prop diameter, and also can have problems with nose-down trim change from the high thrust line when you add power, which then tends to limit just how much power you can install before the plane insists on nosing over on takeoff unless you open the throttle gradually. These pushers do have the advantage of having the tail immersed in the propwash, so arranging the thrustline to optimize the propwash's interaction with the tail can help to some extent. We did this very successfully on our Roadkill Series Curtiss-Wright Junior, although you still have to be a bit careful with it at the start of takeoff run on a relatively rough surface. One thing that can help in these cases is the use of tailwheel landing gear (such as the Junior's), rigged with a fairly high fuselage angle when in the three-point attitude. This moves the motor aft relative to the C/G when in the three-point attitude, which helps hold the tail down until you get enough airspeed for the elevators to become effective.
In most cases the pusher installation does not provide as much propwash over the elevators as a tractor installation. Even if the elevators are immediately in front of the prop, they still don't get as much local airspeed during takeoff as a tractor would provide. This tends to keep the plane from rotating on takeoff until long after the wing is at flying speed, greatly increasing the required runway length. Once it does rotate, it tends to jump aggressively into the air, making it difficult to get a smooth liftoff. Many pushers, such as the VariEze and the Prescott Pusher suffer from this. The Prescott Pusher (among others) also suffers from the two shortcomings I mentioned above, which resulted in relatively inferior performance in comparison to tractor aircraft with the same payload and power loading.
Although a set of elevators immediately ahead of the prop do not generally get much help from the propwash for rotating the plane for liftoff, they can definitely get blanked by the stagnant flow in front of a windmilling prop on landing. This was one of the quirks of that 4 1/2 ft. Roadkill Series Northrop XB-35 we've been developing. If you try to land with less than about 1/4 throttle, the windmilling props blank the elevators and you have no elevator authority to flare with for touchdown. This tends to be very hard on the landing gear. Diving in at high speed on final approach doesn't seem to have much effect on this; the only reliable fix is to land with the throttle a little open.
On full scale aircraft the FOD ("Foreign Object Damage") issues are much worse for a pusher prop than for a tractor in most cases. Blocks of ice shed from the flying surfaces and the fuselage on aircraft expected to fly in instrument flight conditions tend to be an especially severe problem. So can tire treads shed during takeoff. This is generally less of an issue for models, although stones on the runway can still be a problem.
The lightning strike criteria for pushers (class IV, as opposed to class I for tractors) are also more difficult to deal with (also not generally an issue for models).
On turbine engined aircraft, the heat and corrosive gases in the exhaust can be a serious problem, especially if the prop has to operate in reverse for braking on landing roll or especially for maneuvering on the ground. One of the most serious conditions for typical turbine pushers is when an aircraft uses reverse thrust from the prop to back itself away from the gate at the terminal building.
There are a host of other factors to consider as well, although these are some of the biggies. What matters is that the designer properly considers all of the issues from both the prop's and the airframe's point of view. The net result can go either way, depending on the specifics of the particular aircraft and its mission. I've designed both pushers and tractors, and each was appropriate in its individual case. In general I've found that tractors are a better choice in the majority of cases, although there are specific cases where pushers have an overall advantage. Flying wings, where even small differences in pitch and yaw stability can often be important, are one of the cases that often favor pushers.