We assume there is no friction on the surface of the inclined plane and no wind resistance. We want all our Potential Energy to be converted in Kinetic Energy. We have found in real life that wet ice does a great job of reducing Friction.
The Force to consider is the product of the mass of the object multiplied by the acceleration provided by gravity. This force splits into Component Forces, with one component going into the inclined plane (if there had been friction, it would have been proportional to this component) and the other component going in the direction that the sled is traveling down the inclined plane.
It should make sense that if the actual force is splitting into two component forces, each component Force should be less than the actual Force.
We represent the forces with a Right Triangle. If the angle of the inclined plane is small, we should expect the smallest side of the right triangle to be the acceleration in the direction of travel down the inclined plane.
Example Problem:
Given g=9.800 m/s^2 and an inclined plane angle of 20 degrees. We set the length of the hypotenuse to be 9.800 m/s^2.
For our right triangle, the smallest side is the opposite side.
sine(theta) = opposite/hypotenuse
opposite = sine(theta)*hypotenuse
The acceleration in the direction of slide is 3.352.
The larger side is the adjacent side.
cosine(theta) = adjacent/hypotenuse
adjacent = cos(theta)*hypotenuse
The acceleration into the inclined plane is 9.209.
For the above work, everything calculated was an acceleration. If you need to work with forces, any of those accelerations can be multiplied by the
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