Force and Motion
Why Things Keep Going
Why a Sliding Book Stops
Push a book across a table and let go. It slides a little way and stops. Nothing could be more ordinary, and the obvious conclusion is that motion needs a cause. Things move while something pushes them. Take the push away and they settle back into their natural state, which is rest.
That conclusion stood for about two thousand years, and it has the situation exactly backward. Nothing is needed to keep the book moving. Something was needed to stop it, and that something was friction, working quietly at the one surface nobody looks at.
It took so long to see because on Earth nothing is ever left alone. Friction and air are always there, always taking something away. To find the real law you have to imagine away a thing that is present in every observation you have ever made. That is a strange kind of mental move, and physics was slow to learn it.
Two ramps are enough to make the move. Roll a ball down one ramp and up another, and it climbs back almost to the height it started from; the smoother the ramps, the closer it comes. Make the second ramp gentler and the ball rolls farther along it, still climbing to the same height. Now lay the second ramp flat. The ball can never get back up to its height, so it never stops. It rolls on forever, in principle.
This is the first of Newton’s three laws of motion. An object left alone keeps moving in a straight line at a steady speed. Rest is not special: it is just the case where that steady speed happens to be zero. Out in space, with almost nothing to rub against, the law shows itself plainly. The Voyager 1 probe has coasted outward with no engine pushing it ever since it swung past Saturn, decades ago, and it is still going, at about 17 kilometers a second. The Sun’s pull, faint that far out, takes only about six meters a second off that each year.
What a Force Actually Does
Once motion needs no upkeep, the job of a force changes completely. A force is not what keeps something going. It is what changes how something is going.
Change can mean speeding up, slowing down or turning, and all three are the same thing in disguise, because a velocity has a direction as well as a size. A car rounding a bend at a steady speed is accelerating, because its direction is changing, and something has to push it sideways to do that. Here the road pushes on the tires. Take that push away, on ice, and the car carries straight on into the hedge, as the first law says.
The same law settles a puzzle about every car on a level road. If motion needs no push, why does a car need its engine just to hold a steady speed? Because it is never left alone either. The air and the road hold it back all the time. The engine turns the wheels, and the road’s grip on the tires pushes the car forward. At a steady speed that push exactly cancels the holding back. A car cruising straight ahead at a constant speed has no overall force on it at all.
The second law puts a number on the change. It measures an object’s motion by its mass times its velocity, a quantity called momentum. The total force on an object is the rate at which its momentum changes: push twice as hard and it changes twice as fast. For a body that keeps its mass and moves at everyday speeds, this is the familiar form, force equals mass times acceleration, and the same push on twice the mass produces half the acceleration.
So mass, in this law, is not weight. It measures reluctance: how hard an object is to talk out of what it is already doing. On the space station, astronauts and their scales fall around Earth together, so everything floats and a scale reads nothing. They find their mass with springs instead. The springs pull them along a rail, and the less they speed up, the more mass they have.
Nothing Pushes Alone
The third law is the easiest of the three to state, and the easiest to misread. Every force comes in a pair, equal in size and opposite in direction. Lean on a wall and the wall pushes back on you exactly as hard. Earth pulls the Moon, and the Moon pulls Earth with precisely the same strength.
The objection is immediate. If every force is matched by an equal and opposite one, how does anything ever move? Because the two forces of a pair never act on the same object. When you push a crate, your push acts on the crate and the crate’s push acts on you. The crate answers only to the forces on the crate. The partner of your push is not among them, so once your push beats the floor’s grip on the crate, it moves. Meanwhile the crate pushes you backward, which is why you have to brace your feet.
Two skaters pushing off from each other show it cleanly. Each feels the same push for the same moment, so each gains the same momentum, in opposite directions. The lighter one has less mass to carry it, so it leaves faster. Add the two momenta, minding their directions, and they come to zero, exactly as before the push.
That is how a rocket moves in empty space. It does not push against the air. It throws mass out of its back very fast, and the thrown mass pushes the rocket forward in return. Air is not needed, and it only gets in the way: a rocket engine gives more thrust in a vacuum than at sea level.
Seen this way, the third law is a rule about momentum. Forces between two bodies move momentum from one to the other and never change the total. Why that should be so goes deeper still, to the fact that the laws of physics are the same in every place.
Everything Falls at the Same Rate
Drop a hammer and a feather together and the hammer wins, which seems to settle the matter. It does not. At the end of a moonwalk, an Apollo astronaut held up a geological hammer and a falcon feather in front of the television camera and let go of both. With no air to hold the feather back, they hit the ground at the same instant.
Why should heavy and light fall together, when gravity plainly pulls harder on the heavy one? Because the heavy one is harder to move by exactly the same factor. Double the mass and you double the pull, but you also double the reluctance, and the two cancel. Every object gets the same acceleration, and its mass drops out of the answer.
That cancellation is stranger than it looks. Nothing says that the quantity setting how hard gravity pulls on an object has to equal the one setting how hard it is to push. They are different ideas. An early test swung pendulums carrying gold, silver, lead, glass, sand, salt, wood, water and wheat, and found the match held to one part in a thousand. Recently a satellite built for this test alone found it holding to within a few parts in 10¹⁵.
Where the Laws Stop
Take that coincidence seriously, and it turns out to be no coincidence. In Einstein’s theory of gravity, general relativity, gravity is not a force at all. A falling apple is doing what the first law says an object does when nothing acts on it: coasting. Mass and energy bend space and time, and in that bent spacetime the fall is the straightest path there is. What you feel as weight is the ground pushing you off that path.
Speed is the second edge. By the second law, a steady push should add speed without limit. Near the speed of light it stops doing so. The push still adds momentum as steadily as ever, and energy with it, but momentum there is no longer just mass times velocity: it grows without end while the speed barely moves. The mass itself does not grow. It is speed that has a ceiling, and nothing with mass ever reaches the speed of light. The Large Hadron Collider pushed protons to within about three meters per second of it, and no closer.
Size is the third. Inside atoms the idea of a definite path falls apart. An electron travels as a wave, spread out, and is caught whole in one spot only when something records where it is. A wave has no exact position and exact speed at once, so there is no path for a force to bend. A force reshapes the wave instead, and the wave sets the chance of finding the electron at each place.
Newton Is Still Right Where He Worked
None of this makes Newton wrong where he was working. The Apollo crews were steered to the Moon and home again by his laws of motion and gravity. Build a bridge, an elevator, a bicycle or a trebuchet, and they are all you need. Physics did not throw mechanics away. It found the edges of the room mechanics lives in, and then walked out through them.
The habit of mind that found these laws reaches further than the laws do. The book had been stopping for as long as anyone had watched it. The insight came from asking what would happen if the thing that is always there were taken away. Imagining away the constant background to find the law underneath it turns up again and again in physics, and it is close to the whole trick.
