Discussion on Newton's Laws

This subsection draws out several important consequences of the three laws.

1. Newton's laws are vector laws. $\vec F=m\vec a$ is really three independent scalar equations (one per Cartesian axis); a force along $y$ can never change the acceleration along $x$ or $z$.
2. Instantaneous cause-effect. The acceleration at time $t$ depends only on the force acting at that instant, never on the history of forces that acted earlier — e.g. a bowled cricket ball's acceleration after release depends only on gravity and air drag then acting, not on how fast it was bowled.
3. Force and motion direction can differ. Four illustrative cases:
   • Same direction: an apple falling straight down — both its velocity and the gravitational force point downward.
   • At an angle: the Moon orbiting the Earth — the gravitational force is (roughly) radial while the velocity is tangential, so force and velocity are close to perpendicular.
   • Opposite direction: an object thrown vertically upward — velocity is upward, gravity is downward, decelerating it.
   • Zero force, nonzero velocity: a raindrop that has reached terminal velocity (upward air drag exactly balances downward gravity) continues falling at constant speed under zero net force.
4. Superposition of forces. If several forces act together, the acceleration is governed by their vector sum: $\vec F_{net}=\vec F_1+\vec F_2+\cdots+\vec F_n$ — illustrated by the two string tensions of a drawn bow, which add vectorially to the single net force accelerating the arrow.
5. Second-order differential form. Since $\vec a=d^2\vec r/dt^2$, the second law is equivalently $\vec F=m\,d^2\vec r/dt^2$, relating force to the second derivative of the position vector. …