Special cases, freefall and pitfalls to avoid

Free fall: an important special case

In free fall, only gravity acts on the object (air resistance is neglected). The second law then gives:

P = m × a -> m × g = m × a -> a = g

The acceleration is therefore equal to g (approximately 9.8 m/s²), and, crucially, it does not depend on the mass of the object! An iron ball and a feather fall at the same speed in a vacuum, contrary to what one might intuitively expect.

The inclined plane

On a frictionless inclined plane making an angle alpha with the horizontal, the component of the weight parallel to the plane is m × g × sin(alpha). Projecting Newton’s second law onto this axis gives:

m × g × sin(alpha) = m × a -> a = g × sin(alpha)

Here again, mass disappears from the final expression for acceleration.

Summary table

Situation	Acceleration
Free fall	a = g
Frictionless inclined plane	a = g × sin(alpha)
Force F alone, without friction	a = F/m

Common pitfalls to avoid

• Confusing weight (force, in N) and mass (in kg): weight depends on g, mass does not.
• Forgetting that Newton’s second law only applies in a Galilean (non-accelerated) reference frame.
• Believing that a heavier object falls faster: this is incorrect in the absence of friction; the acceleration of free fall does not depend on m.
• Forgetting to correctly project the forces onto the axes when the motion is not purely horizontal or vertical.

These pitfalls are common in exams: clearly identifying the reference frame and drawing a clear diagram will help you avoid them.