Newton's first law

Newton's first law: an object with zero net force on it keeps the velocity it has. At rest, it stays at rest. Moving, it keeps moving in a straight line at constant speed. In symbols: if $\vec{F}_{\text{net}} = \vec{0}$, then $\vec{v}$ is constant.

Two cases, one rule. The rule reads both ways: zero net force gives constant velocity, and constant velocity (zero included) means zero net force. Same fact, two angles.

This contradicts the everyday intuition that a moving object needs something to push it along. The intuition is tuned to a world where friction and air resistance are always present, slowing everything down. Strip those away and the first law shows up plainly: a puck on smooth ice, a spacecraft in vacuum, a hockey stone after the stick has left it. Each carries on at constant velocity with no force in the direction of motion.

The first law holds in inertial reference frames: frames at rest or moving at constant velocity relative to the fixed stars. In an accelerating frame (a braking car, a turning bus, a spinning turntable), the first law looks like it fails and you need fictitious forces to patch it up. Stick to inertial frames in this course and the first law is exact.

The condition $\vec{F}_{\text{net}} = \vec{0}$ covers two situations that look nothing alike.

Static equilibrium. The object is at rest and stays there. A book on a table, a hanging lamp, a parked car. Velocity zero, net force zero.

Dynamic equilibrium. The object moves at constant velocity in a straight line. A car at cruise control on a highway, a glider on a horizontal air track, a skydiver at terminal velocity. Velocity non-zero but constant; net force still zero.

The two cases look different but the condition is the same. Both require zero net force, and the first law treats them as the same case. Cruise-control car: the road's static friction on the driven tires points forward, while air drag and rolling resistance point backward, same magnitude. Sum on the car's diagram is zero; the car coasts at fixed speed. Book on the table: weight down, normal force up. Sum is zero; the book stays still. The arithmetic is the same.

The takeaway: don't read the forces off the velocity. Look at the FBD. If the arrows sum to zero, the object is in equilibrium: at rest or moving.

The most common first-law mistake reverses the rule. The misconception: a moving object must have a force pushing it in the direction of motion, or else it would slow down. This is the Aristotelian view; the first law contradicts it.

Consider a hockey puck coasting east across smooth ice well after the stick has left contact. List the forces. The Earth pulls the puck down: weight. The ice pushes the puck up: normal force. That is the whole list. No object is in contact with the puck horizontally; no field acts on it sideways. The two vertical forces cancel, the horizontal net force is zero, and the puck continues at constant velocity. There is no "force of motion," no "leftover push," no "carried impetus." The puck moves because nothing is stopping it.

The everyday version is misleading because of friction. A book pushed across a table slows down, and the intuition reads that as the natural state. It isn't. The slowing comes from a real backward force: kinetic friction with the table. Remove the friction and the book carries on at constant velocity, the way the first law predicts. The intuition treats friction as invisible because it is always there.

The rule is sharper than "motion needs a force." Motion needs a net force only to change. Constant velocity, including constant zero velocity, needs zero net force. Forces may still act, but they cancel.

Setup. A 0.17 kg hockey puck slides east across frictionless ice at 6 m/s, well after the stick has left contact. Draw the free-body diagram. List the forces. Find the net force. Verify the first law.

List the interactions. Ask which objects are touching the puck and which fields act on it. The Earth pulls down (weight); the ice pushes up (normal force). The stick is no longer in contact, so it exerts no force. No other object touches the puck. That is the full list.

Quantify the forces. Weight: $W = mg = (0.17)(10) = 1.7$ N, downward. Normal force: $N = 1.7$ N upward (the puck doesn't accelerate vertically, so the normal force matches gravity). The free-body diagram has two arrows: one down, one up, both 1.7 N long.

Find the net force. Add the arrows. Vertical components: $N - W = 1.7 - 1.7 = 0$ N. Horizontal components: zero (no horizontal force acts). The net force is the zero vector.

Verify the first law. Zero net force, so $\vec{a} = \vec{0}$, so $\vec{v}$ is constant. The puck continues east at 6 m/s indefinitely. That matches the setup.

The trap to avoid. A common wrong move is to add an eastward arrow labeled "force of motion" or "force of the original strike" to keep the puck going. Every arrow on a free-body diagram has to name an agent. There is no eastward agent here. The stick is gone; the ice supplies no horizontal push. The puck moves not because something pushes it, but because nothing stops it.

The second mistake collapses the net force vector into the velocity vector. The two are independent. Newton's second law ties net force to acceleration ($\vec{F}_{\text{net}} = m\vec{a}$), not to velocity. Acceleration and velocity can point any direction relative to each other; the net force tracks acceleration, not velocity.

Ball at the apex. A ball is thrown straight up. At the highest point of its flight, the ball's velocity is momentarily zero. Its acceleration is not. Gravity has been pulling down through the whole flight, and that doesn't pause when the velocity reaches zero. The net force at the apex is $mg$ downward, the same as at every other point of the flight. The velocity passes through zero; the acceleration does not.

Projectile mid-arc. A baseball is hit upward and to the right. While the ball is on its parabolic arc, the velocity is tangent to the parabola at every point; it changes direction continuously. The net force does not. With only gravity acting (ignoring air resistance), the net force is $mg$ straight down at every moment. Velocity along the tangent, net force straight down.

Constant-speed circular motion. A car drives around a flat circular track at constant speed. The speed is fixed but the velocity is changing direction, so the acceleration is non-zero: centripetal, pointing toward the center. The net force points the same way: toward the center, perpendicular to the velocity.

In each case the misconception lines the net force up with the velocity. It doesn't. The net force lines up with the acceleration. Whenever the acceleration is changing the velocity's direction, or producing a velocity from zero, the two vectors point different ways. The flip side of the first law is the working tool: if the net force is zero, the velocity is constant; if the velocity is not constant, the net force is not zero, and it points the way the velocity is changing.