I'm currently a high school student trying to build an ultralight aircraft from scratch (I'm self taught and still learning from educational videos), but I don't have much savings (about RM 2.5k, which converts to roughly 587 USD), so I could only afford a small engine.
Is it possible to achieve a certain propeller's efficient rpm by using gears while still using a small engine? Like those gearboxes found in cars to increase rpm.
The short answer is yes you can use some cheaply available engines to achieve flight however you are presumably asking about things like small car engines or motorcycle engines etc. The issue you have here is that these engines are inherently different from a use case standpoint than your typical aircraft engine. Generally speaking, most car/motorcycle engines rev too high to be used in an aircraft, you actually want to lower the RPM, not raise it. You can read up on it here and here and here. You can pull it off with a stepper gear box (like a turbo prop uses) and it was even been done on a piston plane back in the 80's by Mooney. The other problem you will have with car engines is cooling them, generally cars are watercooled, something that adds a lot of weight and is not ideal for flight. A two stroke or other similar air cooled motorcycle/scooter engine may be a better thing to use.
Many small engines will produce more than enough horsepower to achieve flight. Keep in mind the early piper cubs only had 40HP which can be easily found in many 2 stroke engines.
Your best bet may be to look for a used VW bug engine as they are very similar to aircraft engines and even were used in some air frames at one point in history. Due to the cars high production numbers the engines are fairly easy to come by.
You did not list a jurisdiction in your question but the regulations and legislation on this change from place to place, so you may need to see what exactly you are allowed to do in your jurisdiction in relation to home built and experimental aircraft.
In order to minimize engine power, reduce speed. The power need of an aircraft scales with the third power of speed, and many designs of the 1920s successfully used motorcycle engines. A high aspect ratio and a low wing loading, like in a glider, will help to get airborne with just 25 HP. Successful designs with such small engines were the Daimler L 20 or the Messerschmitt M 17.
Messerschmitt M 17 (picture source)
Gliders fly without an engine and they do well :) Where I come from, it is common (and legal) to use an old car engine and build it into your own UL plane. Consider getting a car that is very old (70's or 80's) - these can be usually find either for free (after a rear crash) or very very cheap.
These old engines are also easy to remove and fit into an aircraft. In Europe, old Trabant engines are very common for this, but I am sure each region has some car model that is old, unused and waiting in planty in old scrapeyards.
If your legislation allows building a non-certified motor into your plane, than this wold be the way to go. These engines can easily output 40 to 80 kWs of power which is way more than you need.
Safety notice: If you do this, the #1 rule is: Always remember the engine is non-certified and old, so always fly it like it should break the next minute - avoid flying over large forests and water areas.
The answer is yes and no. Can an airplane be designed to fly on very low power? Yes. The Wright 1903 flyer flew on a mere 12 hp gasoline engine. The real question is not can you fly on a lower power engine but can a lower power engine provide a meaningful flight envelope while carrying a realistic useful load over a wide range of atmospheric conditions.
It all depends upon your performance requirements which you need to decide upon before engine selection, but the suggestion of using a VW Beetle engine is a good one. Rotax's line of piston engines have also been a good choice, especially for LSA aircraft.
Actually, you'd be gearing the engine down, not up. Propellers become very inefficient if the tips go supersonic (as well as extremely noisy). You'd want to keep the prop at around 1800-2200 rpm, depending on the size of the prop. Smaller engines typically put out peak power at much higher rpm's than that, so they have to be geared down to match the prop to the engine's power curve.
A typical ultralight engine is the Rotax 582. Designed for ultralight aircraft use, it can sustain high power output for long periods of time. That is current production, so it can be a bit expensive.
Now out of production but they can be found on the used market fairly inexpensively, is the Rotax 503. It has been a very popular ultralight engine.
Motorcycle engines may have less than desirable reliability. Typically, they are designed for brief periods of acceleration, with long periods of cruising at low power output. If you take a stock small motorcycle engine and run it wide open for long periods of time, the cylinder head will probably overheat, blowing the gasket and leaving you with no power. When that happens, you are going to come down, regardless of whether there is a safe place to land, or not.
I have seen Rotax 503's with a gearbox for ultralight use, in the USD 1500 range. More than a small motorcycle engine, but cheaper than a funeral...
Keep in mind that I'm very interested in airplanes, but I don't fly anything full scale, so others may have better advice.
There's a very simple, very light homebuilt called the Sky Pup. I'm not sure, but I think the plans are still being sold. Many of these have been built, and they have a good reputation. I've read that the guys who designed it were aero engineers. The materials are mostly wood and extruded polystyrene foam. The instructions tell you that anything more than 20 hp is too much. You can find out more at machnone.com (My only connection with the Sky Pup is that I bought a set of plans and have read about it here and there.) If you weigh less than 170 lbs, you might find this a good choice. Reduction drives are common, especially with two stroke engines. Another approach is to use half of a VW engine, although some of the power will be wasted. These engines are much too heavy for a Sky Pup, though. There's a design called the Legal Eagle, though, which uses a 1/2 VW engine.
New paramotor engines are probably far beyond your budget, but they might give you an idea of what can be done. They all have belt drives to reduce the propeller rpm. However, keep in mind that reduction drives can have big problems, unless they're engineered just right or have been tested and modified many times.
Unless you are very good at scrounging, or prices are MUCH lower where you are than they are here in the USA, your budget won't go far enough. Maybe there are some glider designs, though, that could be done with less than $1000 (US).
Depending on how an "ultralight" is defined in your country, there may be a bunch of other designs that you can use. For instance, the Pietenpol Sky Scout, which is smaller than the famous Air Camper, is meant for 20 or 30 hp (not sure which). It wasn't known for climbing quickly, but I expect a modern, lighter engine would help with that. I mention it because it's an old, low tech design which may be cheaper to build.
There are MANY other choices out there. Whatever you choose, get some flight training before flying solo. And be really careful when you're building it.
