Newton's second law
This is the law that turns forces into motion. It tells you exactly how much an object speeds up (or slows, or turns) when you push it. Push harder and it accelerates more; make the object heavier and it accelerates less. The picture: the same kick sends a soccer ball flying but barely moves a bowling ball.
Newton's second law says the net force on an object equals its mass times its acceleration: F_net = m a. Both F_net and a are vectors pointing the same way, so the acceleration is always in the direction of the net force. Rearranged, a = F_net / m. The newton is defined from this: 1 N = 1 kg times 1 m/s^2. More generally, force equals the rate of change of momentum, F_net = change in (m v) divided by change in time.
This single equation is the engine of mechanics: give it the forces, and it predicts the motion. Note that it uses the net force, the vector sum of all forces, not any one force alone. And it connects to the first law: if F_net = 0 then a = 0, so the velocity stays constant.
A net force of 10 N on a 2 kg cart gives a = 10 / 2 = 5 m/s^2. The same 10 N on a 5 kg cart gives only 2 m/s^2: the heavier cart accelerates less.
For a fixed force, acceleration falls as mass rises: a = F_net / m.
The m in F = m a is the object's total mass and the F is the net force. Plugging in just one of several forces, or forgetting to add forces as vectors, is the most common mistake.