Unit 24 · Physics: Motion and Forces

A cross-country runner finishes a 5 km race and ends up 300 m from where she started, because the course loops around a park. Two different numbers describe the same run. The 5 km is her distance, the total length of the path. The 300 m, together with the direction back to the start, is her displacement, the straight-line change in position. Distance is a scalar: a size with a unit and nothing more. Displacement is a vector: a size plus a direction.

Physics keeps the two kinds of quantity apart because they add differently. Walk 3 m east and then 4 m north. Your distance is 7 m, but your displacement is 5 m toward the northeast, the hypotenuse of a right triangle. Vectors are drawn as arrows. The length of the arrow shows the size, called the magnitude, and the arrow points in the direction. To add vectors you place them head to tail and draw a new arrow from the first tail to the last head.

A vector can also be split apart. A wind blowing from the southwest can be described as one part blowing east and one part blowing north. These pieces are called components, and they let you handle each direction with ordinary arithmetic. Most of this chapter uses that trick: break a problem into horizontal and vertical components, solve each one, then combine the answers.

Speed and velocity work the same way. Speed is a scalar, the rate at which distance grows. Velocity is a vector, the rate at which displacement changes. A car circling a track at a steady 30 m/s has constant speed but constantly changing velocity, because its direction keeps turning. That distinction sounds fussy, but it is the reason the car needs a force from the road to stay on the curve.

A Metra train pulls out of Union Station. For the first minute its position changes slowly, then faster, then at a steady rate. Plot its position against time and you get a curve that bends upward and then straightens into a line. That picture holds every fact about the trip. The slope of a position-time graph, rise over run, is the velocity. A steep line means fast, a flat line means stopped, and a line sloping downward means moving back toward the start.

A curved position-time graph means the velocity is changing, which is acceleration. Acceleration is the rate at which velocity changes, measured in meters per second per second, written m/s². If a car’s velocity goes from 0 to 20 m/s in 10 s, it gained 2 m/s every second, so its acceleration was 2 m/s². Slowing down is also acceleration, with the acceleration vector pointing opposite the motion. Physicists avoid the word deceleration because the sign already carries that information.

A velocity-time graph makes acceleration easy to see. Its slope is the acceleration, so a straight sloping line means constant acceleration and a horizontal line means constant velocity. This graph hides a second gift. The area between the line and the time axis equals the displacement. A car moving at 15 m/s for 4 s sweeps out a rectangle 15 high and 4 wide, so it moves 60 m. A triangle under a sloping line gives the displacement during steady speeding up.

The three graphs, position, velocity and acceleration, describe the same motion in three ways, and each one is the slope of the one above it. Learning to move between them is the core skill of kinematics, the study of how things move without asking why. Galileo’s ramp data was a position-time table. Its distance grew with the square of time, which is exactly the curve produced by constant acceleration.

A pilot needs to know whether a runway is long enough. A driver needs to know how far a car travels while braking. Both questions have the same shape: the motion has constant acceleration, and some quantities are known while others are not. For that case, five quantities matter: initial velocity, final velocity, acceleration, time and displacement. The kinematic equations link them.

The first equation comes straight from the definition of acceleration. Final velocity equals initial velocity plus acceleration times time, or v = v0 + at. The second comes from the area under the velocity-time graph. Displacement equals initial velocity times time plus one half of acceleration times time squared, or d = v0 t + (1/2) a t². Notice the t squared. That is Galileo’s rule, distance growing with the square of time, now written as algebra.

A third equation removes time entirely, which is useful when you know how far something moved but not how long it took. Final velocity squared equals initial velocity squared plus two times acceleration times displacement, or v² = v0² + 2ad. A car braking at 6 m/s² from 24 m/s stops in a distance of 24 squared over twice 6, which is 48 m. Double the starting speed and the stopping distance quadruples, because the speed is squared.

The most important use of these equations is free fall. Near Earth’s surface every object dropped in a vacuum accelerates downward at about 9.8 m/s², a value written as g. Air resistance changes this for feathers and parachutes, but for a dense ball dropped a few meters it barely matters. A rock dropped from a bridge falls 4.9 m in the first second and about 44 m by the end of the third. Timing the splash is a way to measure a bridge’s height.

Stand at the edge of a table and, at the same instant, drop one coin and flick a second coin sideways off the edge. Listen. They hit the floor together. The flicked coin traveled farther, but it did not take longer to fall. That small experiment holds the whole theory of projectiles. The horizontal and vertical motions are independent. Gravity pulls down and only down, so it has no effect on sideways speed.

A projectile is any object moving only under gravity once it has been launched: a thrown ball, a jumping frog, a stream of water from a hose. Split its velocity into components. Horizontally there is no force, so the horizontal velocity stays constant and horizontal distance is just speed times time. Vertically the object is in free fall. It slows as it rises, stops for an instant at the top, then speeds up on the way down, all at 9.8 m/s².

Putting the two together produces the curve called a parabola. The shape is symmetric if the launch and landing heights are equal: the time up equals the time down, and the landing speed equals the launch speed. The distance a projectile travels horizontally is its range. In a vacuum the range is greatest at a launch angle of 45 degrees, and any two angles that add to 90 degrees, such as 30 and 60, give the same range.

Air changes the picture for real objects. A baseball at Wrigley Field feels air resistance, and on windy days the flags along the outfield tell fielders which way to shade. But the coin experiment still works, and so does the physics of a dropped package from a rescue plane: it keeps the plane’s forward speed while it falls, so the pilot must release it well before reaching the target.

Swing a ball on a string around your head and let go. The ball does not fly outward. It flies off along a straight line, tangent to the circle, in whatever direction it was moving at the instant of release. Nothing was pulling it outward. The string was pulling it inward the whole time, and that inward pull was the only reason it went in a circle.

Velocity is a vector, so a change of direction is a change of velocity, and a change of velocity is an acceleration. An object moving at steady speed v around a circle of radius r has an acceleration of v² divided by r, always pointing toward the center. This is centripetal acceleration, from Latin for center-seeking. It grows with the square of the speed, so a curve taken twice as fast needs four times the inward acceleration.

Newton’s second law says every acceleration needs a force, so something must push or pull toward the center. For a ball on a string it is tension. For a car on a flat curve it is friction from the road. For the Moon it is gravity. The term centripetal force is not a new kind of force; it is a job description filled by whichever real force is available. If that force cannot keep up, the object drifts outward along a tangent, which is what a skidding car does.

The feeling of being flung outward on a spinning ride comes from your body trying to go straight while the seat curves in under you. The same effect shows up on a lab turntable: a coin near the rim needs more friction to stay put than a coin near the center, because at the same rotation rate its speed is higher. The sharpest curves on Chicago’s elevated tracks are marked with low speed limits for exactly this reason.

A hockey puck slides across ice and keeps going. The same puck on a carpet stops in a meter. For two thousand years the carpet seemed like the normal case: things stop unless you keep pushing, so motion must need a cause. Galileo saw it the other way. The puck on carpet stops because friction acts on it. Remove the friction and it would slide forever. Motion does not need a cause; changes in motion do.

Newton made that idea his first law. An object at rest stays at rest, and an object in motion keeps moving at constant velocity, unless a net force acts on it. The net force is the vector sum of all the forces on an object. A book on a table has two forces on it, gravity down and the table pushing up, and they cancel. The net force is zero, so the book’s velocity stays what it is, which is zero.

The property that resists changes in motion is inertia, and mass measures it. A bowling ball and a soccer ball can sit equally still, but try to push each one into motion and the difference is obvious. In a spacecraft, where nothing weighs anything, a large steel tool is still hard to start moving and hard to stop. Inertia belongs to mass, not to weight.

The first law explains the lurch you feel when a bus starts. Your body was at rest and it tries to stay at rest, so the seat back has to push you forward. When the bus brakes, your body keeps moving forward at the old speed until something stops it. That something is either a seat belt, or the dashboard. The last section of this chapter returns to that choice.

Push an empty shopping cart and it leaps forward. Load it with forty pounds of groceries and the same push barely gets it rolling. Newton’s second law says exactly how much difference the load makes. The acceleration of an object equals the net force on it divided by its mass, or a = F/m, more often written F = ma. Double the force and the acceleration doubles; double the mass and it halves.

The unit of force is the newton, symbol N. One newton is the force that gives a 1 kg mass an acceleration of 1 m/s². It is about the weight of a small apple. A 1,200 kg car accelerating at 3 m/s² needs a net force of 3,600 N from the road. The word net matters. Engines, friction and air drag all push on the car; only their combined vector sum shows up in F = ma.

The second law also links mass to weight, which are not the same thing. Mass is the amount of matter and its inertia, measured in kilograms, and it is the same everywhere. Weight is the force of gravity on that mass, in newtons, equal to mg. A 70 kg person weighs about 686 N on Earth and about 112 N on the Moon, where g is 1.6 m/s², yet has the same 70 kg of mass in both places.

The second law is a vector equation. The acceleration always points in the direction of the net force, not necessarily in the direction of motion. A ball thrown upward moves up while gravity, and therefore its acceleration, point down. That is why it slows. Every problem in the rest of this chapter follows one recipe: find all the forces, add them as vectors to get the net force, and divide by mass.

Lean against a wall and the wall leans back on you. That is not a figure of speech. Newton’s third law says that when object A exerts a force on object B, then B exerts a force of equal size and opposite direction on A. The forces always come as a pair. You cannot push on something without being pushed back, and the two pushes are equal, no matter how different the objects are.

This leads to a famous puzzle. If a horse pulls a cart forward, and the cart pulls the horse backward with an equal force, how does anything move? The answer is that the two forces act on different objects. The forward pull acts on the cart and the backward pull acts on the horse. To know whether the cart accelerates, you add up only the forces on the cart: the horse’s pull forward and friction backward. The pair forces never cancel, because they are never on the same object.

The tool for keeping this straight is a free-body diagram. Draw the object as a dot. Draw every force acting on that object as an arrow pointing away from the dot, labeled with its cause: gravity, the normal force from a surface, tension in a rope, friction, a push. Leave out the forces the object exerts on other things. Then add the arrows as vectors. The sum is the net force, and F = ma finishes the job.

A free-body diagram of a person standing in an elevator shows two forces: weight down and the floor’s normal force up. If the elevator accelerates upward, the normal force must be larger than the weight, which is why you feel heavier. If the cable snapped, the normal force would drop to zero and you would feel weightless while both you and the elevator fell together. Astronauts on the space station feel that every minute of every day.

A heavy box sits on a floor. Push gently and nothing happens. Push harder and still nothing. Push harder still and the box suddenly breaks free, and now it takes less force to keep it moving than it took to start it. This everyday experience contains everything important about friction. Friction is the force between two touching surfaces that resists their sliding past each other.

While the box is still, static friction matches your push exactly, up to a limit. That limit equals a number called the coefficient of static friction times the normal force pressing the surfaces together. Once the box slides, kinetic friction takes over, and it is usually smaller, so the box lurches. Kinetic friction is roughly constant regardless of speed. Neither kind depends much on the area of contact, which surprises most people.

The coefficient of friction, written with the Greek letter mu, is a number with no unit that describes the pair of surfaces. Rubber on dry concrete has a coefficient near 1, which is why tires grip. Waxed skis on snow are near 0.1. Ice on ice is lower still. Wet leaves, oil and loose gravel all lower the coefficient between tire and road, which is the physics behind every slippery-when-wet sign.

Friction is not simply the enemy. Without it you could not walk, a car could not start or turn, and a nail would slide out of the wood. Engineers spend as much effort adding friction, with treads, textured floors and brake pads, as removing it with oil, ball bearings and smooth ice. The rule of thumb is that friction does what the situation demands, up to its limit, and problems begin when that limit is reached.

A wheelchair ramp, a loading dock, a sledding hill, Galileo’s groove: all are inclines, and all are solved the same way. Gravity pulls straight down, but the object can only move along the slope. So tilt your axes. Let one axis run along the surface and the other point perpendicular to it. Now the weight splits into two components, and each one has a clear job.

The component along the slope is mg times the sine of the angle. It is the part of gravity that tries to slide the object downhill. The component into the slope is mg times the cosine of the angle. The surface pushes back with a normal force equal to it, and friction depends on that normal force. On a flat floor the angle is zero, the sine is zero and nothing slides. On a vertical wall the sine is one and the full weight pulls down.

Galileo’s ramp was gentle, so the sine was small and the ball’s acceleration was a small fraction of g. That is precisely why it worked. Without friction, acceleration along a frictionless incline is g times the sine of the angle. At 30 degrees the sine is 0.5, so the acceleration is 4.9 m/s². Steeper ramps give larger accelerations, all the way to free fall at 90 degrees.

Friction adds a third arrow to the free-body diagram, pointing up the slope for an object sliding down. A box will stay put as long as the downhill component of gravity does not exceed the maximum static friction. Since both grow and shrink with the same mass, the mass cancels: the angle at which a box starts to slide depends only on the coefficient of friction. Road engineers use this in setting the grade of highway ramps and the banking of curves.

A car traveling at 15 m/s, about 34 miles per hour, hits a wall. The car stops in a tenth of a second, its front end folding. A passenger without a seat belt does not stop, because nothing has pushed on her yet. Newton’s first law carries her forward at 15 m/s across the space of the cabin until the windshield or dashboard applies the force the car did not. That second collision is the one that injures.

The size of that force follows from the kinematic equations and the second law. Stopping from 15 m/s over 0.01 m, the thickness of a hard surface’s give, requires an acceleration of over 11,000 m/s², more than a thousand times g. Stopping over 0.5 m, the stretch of a seat belt and the crush of a crumple zone, requires about 225 m/s², roughly 23 g. Same speed, same passenger, same change in velocity. Fifty times less force.

Every safety device in a car is a way to make the stop take longer. The crumple zone folds so that the cabin decelerates over a greater distance. The seat belt stretches slightly and spreads the force across the strong bones of the hips and shoulder. The air bag inflates in a few hundredths of a second and then deflates while the head sinks into it, extending the stopping time still further. Together they trade a short, violent stop for a longer, gentler one.

Crash-test data have been the evidence behind decades of design changes. The National Highway Traffic Safety Administration reports that lap and shoulder belts reduce the risk of death for front-seat occupants by roughly 45 percent in passenger cars. The physics is the same for a cyclist’s helmet, a gymnast’s mat and a ship’s fender: spread the stop over more distance and time, and the force falls.

A bowling ball rolling slowly and a tennis ball flying fast can be equally hard to stop. What they share is momentum, the product of mass and velocity, written p = mv. Momentum is a vector that points the way the velocity points. Its unit is the kilogram meter per second. A 0.15 kg baseball at 40 m/s carries 6 kg·m/s; a 1,500 kg car at 30 m/s carries 45,000 kg·m/s.

Newton actually wrote his second law in terms of momentum: the net force on an object equals the rate at which its momentum changes. Multiply both sides by the time the force acts and you get impulse. Impulse equals force times time, and it equals the change in momentum. A big force for a short time and a small force for a long time can produce the same change in momentum. That single sentence is the physics of every catch, crash and cushion.

A baseball catcher pulls his glove back as the ball arrives. The ball’s momentum change is fixed by its speed and mass, but by letting the stop take longer, the catcher lowers the force on his hand. A boxer rolls with a punch for the same reason. The seat belt from the last chapter is an impulse device: same momentum change, longer time, smaller force. The area under a force-time graph is the impulse, just as the area under a velocity-time graph is displacement.

Impulse also runs the other way. A rocket engine delivers force over a long burn, and the momentum the ship gains is the whole area under that force-time curve. The Apollo 13 crew needed a precise change in momentum to reach Earth. They got it by burning a known engine for a timed number of seconds, and the stopwatch mattered as much as the engine.

Two skaters stand still on smooth ice, facing each other. One pushes the other. Both move apart, and the lighter skater moves faster. Before the push the total momentum was zero. After it, one skater’s momentum points left and the other’s points right, and the two add to zero again. The push was an internal force, a third-law pair inside the system, and internal forces cannot change the total momentum.

That is the law of conservation of momentum. For any system of objects, if the net external force is zero, the total momentum before an event equals the total momentum after it. The reason is Newton’s third law. Every force one part exerts on another is matched by an equal and opposite force back. Each pair delivers equal and opposite impulses, so the changes in momentum cancel within the system.

The law makes hard problems easy. A 2 kg cart at 3 m/s hits a 1 kg cart at rest and the two stick together. Total momentum before: 6 kg·m/s. After, the combined 3 kg mass must carry the same 6 kg·m/s, so it moves at 2 m/s. You never needed to know the force between the carts or how long it acted. Momentum bookkeeping skips the messy middle of the collision entirely.

Recoil is the same law seen from the other side. A rifle kicks backward when the bullet goes forward. A rocket moves forward because it throws exhaust backward at high speed, which is why rockets work in empty space with nothing to push against. Every burn of Aquarius’s descent engine was a momentum trade: hot gas one way, spacecraft the other, and the sum unchanged.

On a pool table the cue ball strikes a stationary ball at an angle. The two balls leave in different directions, and neither one goes the way the cue ball was going. Yet if you add their momentum vectors, head to tail, the sum points exactly along the cue ball’s original path with the same length. Momentum is conserved as a vector, so it is conserved along the x axis and the y axis separately.

That makes two-dimensional collisions a matter of bookkeeping in two columns. Write the total x-momentum before and after; they must match. Do the same for y. If a 2 kg puck moving east at 3 m/s hits a puck at rest and afterward moves northeast, the pucks’ combined northward momentum must be zero, so the second puck has to move somewhat southward. Crash investigators use exactly this reasoning, working backward from skid marks to the speeds before impact.

Collisions come in kinds. In an elastic collision the total kinetic energy after equals the total before. Billiard balls and colliding steel bearings come close. In an inelastic collision some kinetic energy turns into heat, sound and deformation. Momentum is still conserved. If the objects stick together it is perfectly inelastic, and the loss of kinetic energy is the largest possible. Car crashes are inelastic, and the crumpling that eats the energy is by design.

Momentum and energy are different conserved quantities, and mixing them up is the most common error. Two identical clay balls moving toward each other at equal speeds stick and stop. Momentum: zero before, zero after, conserved. Kinetic energy: plenty before, none after, all turned to heat. Nothing was violated. Kinetic energy is only one form of energy, while momentum has no other form to hide in.

Toss a hammer end over end across a room. The head and handle trace wild loops, but one point on the hammer follows a smooth parabola, exactly like a thrown ball. That point is the center of mass, the average position of all the mass in the object. For a uniform ruler it sits at the middle. For a hammer it sits near the heavy head. For a doughnut it sits in the hole, where there is no material at all.

The center of mass obeys a simple rule. Its motion depends only on the external forces on the system and the total mass, as if everything were concentrated there. That is why a fireworks shell that bursts in midair still has a center of mass moving along the original arc, even as fragments fly everywhere. It is also why conservation of momentum can be stated another way: with no external force, the center of mass moves at constant velocity.

Balance depends on the center of mass too. An object resting on a surface tips over when its center of mass moves past the edge of its base of support. Wide stances and low centers of mass are stable; that is why a wrestler crouches and why a tall bus takes corners slowly. A high jumper using the flop technique arches so that her center of mass passes below the bar while her body, piece by piece, passes over it.

You can find the center of mass of a flat shape by hanging it from two different points and drawing a vertical line down from each. The lines cross at the center of mass. In space, a two-body system like the Earth and Moon orbits around their shared center of mass, which lies inside the Earth but well away from its center. Astronomers find unseen planets by watching stars wobble around a center of mass they share with something dark.

Try to open a door by pushing near the hinge. It barely moves. Push at the handle with the same force and it swings easily. The difference is not the force but where it acts. Torque is the turning effect of a force, and it equals the force times the lever arm, the perpendicular distance from the pivot to the line of the force. Torque is measured in newton meters. Long wrenches, long handles and long oars all exist to multiply torque.

A force pointed straight at the pivot has no lever arm and no torque, however large it is. Only the part of the force perpendicular to the lever counts. That is why you pull a wrench at a right angle to its handle and why a bicycle pedal delivers the most torque when the crank is horizontal. Torque, like force, is a vector; by convention a counterclockwise turn is positive and clockwise is negative.

For an object to be in rotational equilibrium, the torques must add to zero, just as forces must for ordinary equilibrium. A seesaw balances when the weight on one side times its distance from the pivot equals the weight on the other side times its distance. A 30 kg child 2 m from the center balances a 60 kg adult sitting 1 m from the center. Cranes, bridges and bookshelf brackets are designed with the same two rules: net force zero, net torque zero.

Your body works by torque. Muscles attach close to joints, so they must pull with large forces to hold modest loads at the hand. Holding a 5 kg bag with the forearm level means the biceps pulls with something like 150 N, because its lever arm is short and the bag’s is long. The reward is that a small muscle contraction moves the hand a long way, fast. Nature traded force for speed.

A figure skater spins with arms out, then pulls them in and spins dramatically faster. No one pushed her. She simply rearranged her own mass. Rotation has its own version of momentum, called angular momentum, and it is conserved when no external torque acts. Angular momentum depends on how fast something spins and on how its mass is spread out around the axis, a quantity called moment of inertia.

Moment of inertia plays the role that mass plays in straight-line motion. Mass far from the axis counts much more than mass near it. Pulling the arms in reduces the skater’s moment of inertia, so to keep angular momentum constant her spin rate must go up. Divers tuck to somersault faster and stretch out to slow their rotation before entering the water. Gymnasts, cats and falling astronauts all do the same.

Angular momentum is also a vector, pointing along the spin axis, and its direction is conserved as well as its size. That is why a spinning top stays up and a moving bicycle stays balanced far more easily than a stopped one. A gyroscope in a spacecraft holds its axis fixed in space, giving Apollo’s guidance system a reference direction while the ship itself turned. A spinning turntable on the lab bench resists being tilted for the same reason.

On the largest scales angular momentum shapes the universe. A slowly turning cloud of gas collapses under gravity into a star, and as its mass draws inward it spins faster, flattening into a disk. Planets form in that disk, all circling the same way, which is why the planets of the solar system orbit in nearly the same plane and the same direction. Collapsed stars called pulsars, no larger than a city, can spin hundreds of times per second for the same reason.

The apple falls; the Moon does not fall away. Newton’s insight was that both are the same event. The Moon is falling toward Earth all the time, but it also moves sideways fast enough that the ground curves away beneath it. Gravity that reaches the top of a tree, he reasoned, should reach the Moon too, though weaker with distance. He worked out how much weaker: the pull drops with the square of the distance.

The law of universal gravitation says the force between any two masses equals G times the first mass times the second, divided by the distance between their centers squared. G is the gravitational constant, about 6.67 times 10 to the minus 11 newton meters squared per kilogram squared. That tiny number is why you do not feel the pull of a nearby building. It takes a planet-sized mass to make the force noticeable.

The law explains why g is 9.8 m/s² on Earth’s surface and 1.6 m/s² on the Moon. Plug in Earth’s mass and radius and out comes 9.8. The acceleration does not depend on the falling object’s mass, because that mass cancels, which is why a hammer and a feather fall together on the airless Moon, as an Apollo 15 astronaut demonstrated on camera in 1971. Double your distance from Earth’s center and the pull drops to one quarter.

Newton could not measure G. In 1798 Henry Cavendish did, with a delicate torsion balance: two small lead balls on a rod suspended by a fine wire, drawn toward two large lead balls. The twist of the wire measured a force smaller than a grain of sand’s weight. With G known, the mass of the Earth followed, and then the mass of the Sun. One measurement in a quiet shed weighed the solar system.

Before anyone knew why planets move, Johannes Kepler worked out how. He inherited decades of careful naked-eye measurements of Mars from the Danish astronomer Tycho Brahe. For years Kepler tried to fit those positions to circles, and the best circle missed by about eight minutes of arc, a small angle but far larger than Tycho’s errors. Kepler trusted the data over the circle and threw the circle out.

His first law says each planet moves on an ellipse, a stretched circle with two special points called foci, and the Sun sits at one focus. Most planetary orbits are nearly circular; Earth’s distance from the Sun varies by only about 3 percent over a year. Mars is more stretched, which is why its orbit gave the circle away. Comets follow extremely stretched ellipses and spend most of their time far from the Sun.

The second law says a line from the Sun to the planet sweeps out equal areas in equal times. Near the Sun the planet moves fast; far away it crawls. This is conservation of angular momentum in disguise. The third law, found ten years later, says the square of a planet’s orbital period is proportional to the cube of its average distance from the Sun. A planet four times farther out takes eight times longer to go around.

Newton showed that all three laws follow from an inverse square force. That was the decisive test of universal gravitation: a single law derived by thinking about falling apples reproduced patterns Kepler had dug out of Tycho’s tables. The third law now works in reverse. Measure how far and how fast a moon orbits a planet, and you can weigh the planet without ever visiting it.

Newton imagined a cannon on a high mountain. Fire the ball slowly and it lands nearby. Fire it faster and it lands farther away. Fire it fast enough and the ground curves away as quickly as the ball falls, so it never lands at all. It circles the Earth and comes back to the cannon. That thought experiment, drawn in a book published after his death, is the whole idea of a satellite. Orbiting is falling with enough sideways speed to keep missing.

The speed needed depends only on the mass of the planet and the distance from its center. For a circular orbit, gravity supplies exactly the centripetal force, and the equation gives the orbital velocity as the square root of G times the planet’s mass divided by the orbital radius. Closer orbits are faster. Just above Earth’s atmosphere a satellite must move at nearly 8 km/s and completes a lap in about 90 minutes. The Moon, 60 times farther out, moves at about 1 km/s and takes a month.

Different jobs need different orbits. Weather and spy satellites fly low for sharp pictures. Navigation satellites sit at medium altitudes. A satellite placed about 35,800 km above the equator takes exactly one day to go around, so it hangs over the same spot on the ground. That is a geostationary orbit, and it is where communication and many weather satellites live, so that a fixed dish can point at them.

To leave a planet entirely you need escape velocity, the speed at which an object’s kinetic energy matches the energy needed to climb out of the planet’s gravity well. For Earth it is about 11.2 km/s from the surface. Apollo 13’s problem was that its ship had that kind of speed and needed to give some of it back precisely, so that Earth’s gravity could capture it into a path that ended in the atmosphere at the right angle.

The International Space Station orbits about 400 km up, moving at roughly 7.7 km/s and circling the Earth about every 90 minutes. Its crew sees around 16 sunrises a day. Inside, tools drift, water forms wobbling spheres and astronauts sleep strapped to the wall. It is tempting to say there is no gravity up there. There is plenty. At 400 km Earth’s pull is still about 90 percent as strong as on the ground.

The astronauts float because everything on the station, people, air, tools and the walls, is in free fall together. Recall the elevator with the snapped cable from the last chapter: the floor no longer pushes up, so you feel weightless even though gravity is pulling as hard as ever. The station is that elevator, falling continuously, but with enough sideways speed that it never reaches the ground. Weightlessness is the feeling of nothing pushing back.

That state, sometimes called microgravity, is why the station is a laboratory. Flames burn as round blue spheres because hot air does not rise. Crystals grow without settling. Bones and muscles, no longer loaded, weaken by measurable amounts each month, so astronauts exercise about two hours a day and are studied carefully on return. Their bodies are experiments in what a lifetime of standing up against gravity really does.

The station is also a lesson in the small forces this unit has ignored. Thin air at 400 km drags on it, and it loses a little altitude every day, so visiting spacecraft periodically boost it back up. The Moon’s gravity and the Sun’s tug on it as well. Even the ISS is not quite the ideal orbit of the equations, and keeping it there is a matter of momentum bookkeeping, done month after month.