AI Tutor
If gravity pulls every falling object with a constant downward force, why doesn't a falling skydiver accelerate infinitely until impact?
As speed climbs, air molecules smash into the falling object at higher rates, creating an opposing resistive force (drag) that balances gravity and caps speed at a finite terminal velocity.
How do we write down acceleration mathematically when the opposing force depends directly on how fast the object moves?

Setting Up Equations of Motion

In 1687, Isaac Newton formalized the relationship between net force and motion in his Principia, showing that net force equals mass times acceleration (Fnet=maF_{\text{net}} = ma).
Depending on whether we want to find velocity as a function of time tt or displacement xx, we express acceleration in one of two standard calculus forms:
a=dvdtora=vdvdxa = \frac{dv}{dt} \quad \text{or} \quad a = v\frac{dv}{dx}
When setting up resistive forces, remember that drag always opposes the direction of instantaneous motion. Linear drag is modelled as R=kvR = kv or R=mkvR = mkv, while quadratic (high-speed) drag is modelled as R=kv2R = kv^2 or R=mkv2R = mkv^2, where k>0k > 0 is a constant of proportionality.
Establishing clear sign conventions is essential before writing equations. For downward motion with downward defined as positive, mdvdt=mgβˆ’kvm\frac{dv}{dt} = mg - kv; for an object projected upward with upward defined as positive, both gravity and drag oppose motion: mdvdt=βˆ’mgβˆ’kvm\frac{dv}{dt} = -mg - kv.

Setting Up Resistive Forces

When setting up resistive forces, remember that drag always opposes the direction of instantaneous motion.