Newton's First Law

Launch a puck at 3.0 m s⁻¹ across three surfaces and measure how quickly it slows. On carpet it loses speed at 2.5 m s⁻² and stops after 1.2 s and 1.8 m; on a polished floor, at 0.30 m s⁻², after 10 s and 15 m; on smooth ice, at 0.050 m s⁻², after 60 s and 90 m. The launch was identical each time, so the launch does not decide how long the motion lasts; the surface does. Galileo argued the same way with a ball rolling down one ramp and up a facing one: the smoother the ramps, the closer it climbs back to its starting height, and if the second ramp is laid flat the ball never regains that height and has no reason to stop. Extrapolate the trend to zero resistance and the stopping time grows without bound. So the carpet's backward push is what stops the puck, not a natural running down of motion, and Newton turned the extrapolation into a law: with no net force, velocity does not change at all.

The first law draws no line between a body at rest and one moving at constant velocity: both have dv/dt = 0, both have ΣF = 0, and both are in equilibrium. A balloon drifting with the wind at a steady 3.0 m s⁻¹ and constant altitude has its buoyancy balancing its weight exactly as it would if it hung motionless in still air. The opposite trap is rest for an instant. A ball tossed straight up stops at the top of its flight, yet 0.10 s earlier it was rising at 0.98 m s⁻¹ and 0.10 s later it is falling at 0.98 m s⁻¹. Its velocity changes by 9.81 m s⁻¹ every second, before, during and after that instant, so it is never in equilibrium. Because equilibrium is about how velocity changes, it can only be judged over an interval: v = 0 at one moment says nothing about the forces, while v unchanged over any stretch of time says they balance.

ΣF = 0 is a vector statement, so in a plane it is two scalar equations. Pick one axis along the motion and one across it, resolve every force, and set each sum to zero. A traveller drags a 15.0 kg suitcase over level ground at a steady 1.2 m s⁻¹ using a strap at 35.0° above the horizontal with a tension of 40.0 N. Along the ground: 40.0 N × cos 35.0° − f = 0, so the ground's friction is f = 32.8 N. Vertically: n + 40.0 N × sin 35.0° − 147 N = 0, where 147 N is the weight, 15.0 kg × 9.81 m s⁻², so the ground pushes up with n = 124 N, less than the weight because the strap carries part of it. The 1.2 m s⁻¹ appears in neither equation; any steady speed gives the same forces. Count unknowns before you start: two equations can fix at most two, so a planar problem hiding three unknown quantities needs a further physical relation, not more algebra.

Inertia is the name for a body's tendency to keep whatever velocity it has, and mass is its measure: the larger the mass, the smaller the change in velocity a given push produces, which the second law will make exact. It is not a force. A hockey stick is in contact with a puck for about 0.01 s and sends it off at 30 m s⁻¹; 0.20 s later the puck is 6.0 m away and still moving at very nearly 30 m s⁻¹, with the stick nowhere near it. Throughout that flight the forces on the puck are its weight, the ice pushing up and a small friction pointing backwards, and not one of them points forwards. Nothing is needed to carry the velocity along. That gives a test for every arrow on a free-body diagram: each force must name the object exerting it, such as the Earth, the ice or a rope. A “force of motion” or “force of inertia” has no such object, which is how you know it does not belong.

Acceleration depends on who is measuring it, so the first law needs a qualifier. Rest a puck on an air-hockey table inside a train that pulls away at 0.50 m s⁻². From the platform the puck stays where it was, as the law predicts, while the carriage slides forward underneath it. From the carriage the puck accelerates towards the rear at 0.50 m s⁻² and has slid 1.0 m back after 2.0 s, although no rope, surface or field pushes it that way. The carriage frame fails the test, so the law is restricted to inertial frames: frames in which a body with zero net force keeps a constant velocity. That is why the first law is more than the second law with the force set to zero; it identifies the frames in which the second law may be used at all. One inertial frame gives infinitely many, because a frame moving at constant velocity V relative to it measures v′ = v − V and therefore the same acceleration, a′ = a.

No laboratory is perfectly inertial, so the useful question is whether its acceleration shows at the precision you need. Earth turns once every 86 164 s, an angular speed ω = 7.29 × 10⁻⁵ rad s⁻¹, so a point on the equator accelerates towards Earth's axis at ω²R = 0.034 m s⁻², about 0.35% of g. For bodies at rest this is already folded into the measured value of g and the direction a plumb line hangs. What remains is a slow sideways drift of moving bodies that grows with speed and time. For a glider crossing an air track in a few seconds it is a fraction of a millimetre, far below friction and timing errors. Over hours or hundreds of kilometres it wins: a long Foucault pendulum at 52° latitude turns its plane of swing by about 12° every hour, and large storms spin in opposite senses in the two hemispheres. Treat the ground as inertial for bench mechanics, knowing that this is an excellent approximation rather than a fact.