Why surfaces grip: static and kinetic friction
Slide a book across a desk and something resists you; let it sit and it stays put even on a gentle slope. Both are friction, the sideways force between surfaces in contact, and it comes in two flavours. Static friction acts while the surfaces are not sliding, matching your push exactly so nothing moves — up to a limit. Kinetic friction acts once the object is sliding, opposing the motion with a roughly steady force.
Notice the story in that graph: as you push harder, static friction quietly grows to keep pace, so the object stays still. At a threshold it can no longer keep up, the object breaks free, and friction abruptly drops — which is why a stuck drawer suddenly lurches once it starts to move.
The two friction laws
Static friction can take any value up to a maximum set by the coefficient of static friction times the normal force.
Kinetic friction has a fixed magnitude, the coefficient of kinetic friction times the normal force.
The coefficient of friction \mu is a dimensionless number capturing how rough the pair of surfaces is — around 0.6 for rubber on dry concrete, near 0.05 for waxed skis on snow. For a given pair, \mu_s is usually a bit larger than \mu_k, which is exactly why it takes more force to start something sliding than to keep it sliding. Both frictions grow with the normal force, which is the deep reason a heavier crate is harder to shove.
Worked example: to push or not to push
A 10 kg box sits on level ground with \mu_s = 0.50 and \mu_k = 0.30. First find the normal force: on flat ground with no vertical acceleration it balances gravity, so N = mg = 10 \times 9.8 = 98\ \text{N}. The most static friction can offer is f_{s,\max} = \mu_s N = 0.50 \times 98 = 49\ \text{N}.
Push with 40 N and nothing happens: static friction simply matches you at 40 N, still below its 49 N ceiling, so the box stays put. Push with 60 N and you exceed the ceiling — the box breaks free and now kinetic friction takes over.
Once sliding, the net force is the push minus kinetic friction, giving the acceleration.
Air resistance and terminal velocity
Air pushes back too. Air resistance, or drag, is a contact force from the fluid you move through, and unlike dry friction it grows rapidly with speed. Drop a skydiver and gravity accelerates her downward, but the faster she falls the harder the air pushes up. Eventually the two balance exactly, the net force vanishes, and — by the first law again — she falls at a constant terminal velocity the rest of the way down.