Jugami Physics

High School · Physics · Unit 1

Motion and Kinematics

How far, how fast, which way, and how quickly that changes: the language physics uses to describe anything that moves.

A sprinter bursting from the blocks, a car braking at a red light, a basketball arcing toward the hoop. Before we ask why things move (that's Newton's laws, in Unit 2), we need a precise way to say how they move. That description is called kinematics.

On this page you'll learn the difference between distance and displacement and between speed and velocity, what acceleration really means, how to read motion graphs, and how to use five equations that solve almost any problem with steady acceleration. Then you'll drop things, throw things, and launch things in a projectile lab, meet Galileo and his rolling balls, and take a quiz.

Start with displacementJump to the quiz
Blueprint sketch of a ball thrown from the ground. Its path is a curved arch called a parabola. At three points along the path, a blue arrow shows the sideways part of its velocity, which stays the same length, and an orange arrow shows the up-and-down part, which points up on the way up, is zero at the top, and points down on the way down. The maximum height H and the range R are marked, and a downward arrow shows gravity, g = 9.8 meters per second squared. H R θ g Up and across are independent motions

On this page

Key words first

Position

Where something is, measured from a chosen starting point (the origin), with a direction. Measured in meters (m).

Distance

The total length of the path you actually traveled. It only adds up and never goes negative.

Displacement

The straight-line change in position from start to finish, with a direction. Walk out and back home: lots of distance, zero displacement.

Speed

How fast you cover distance, in meters per second (m/s). A speedometer shows speed.

Velocity

Speed with a direction: how fast your position changes, and which way. 20 m/s north is a velocity.

Acceleration

How fast your velocity changes, in meters per second per second (m/s²). Speeding up, slowing down and turning are all acceleration.

Idea 1: Distance and displacement

Suppose you walk to a friend's house: 3 blocks east, then 4 blocks north. How far did you go? That depends on what you mean.

The distance is the length of the path your feet followed: 3 + 4 = 7 blocks.

The displacement is the straight arrow from where you started to where you ended up. It forms the long side of a right triangle, so you can find its size with the Pythagorean theorem: √(3² + 4²) = √25 = 5 blocks, pointing toward the northeast (53° north of east, to be exact).

Displacement only cares about the start and the finish. If you walk around the block and come back to your front door, your distance might be 400 m, but your displacement is zero.

displacement = final position − starting position

A map of city streets drawn as a grid. A walker starts at the bottom left, walks 3 blocks east along an orange path, then 4 blocks north. The total distance walked is 7 blocks. A blue arrow goes straight from the start to the finish: the displacement, 5 blocks long, pointing about 53 degrees north of east. 53° Start Finish 3 blocks east 4 blocksnorth displacement 5 blocks, 53° north of east Distance walked: 3 + 4 = 7 blocks Displacement: √(3² + 4²) = 5 blocks, toward the northeast
Walking 3 blocks east and 4 blocks north: the distance is 7 blocks, but the displacement is 5 blocks in a straight line toward the northeast.

Idea 2: Scalars and vectors

Some quantities are completely described by a number and a unit, like "7 blocks" or "20 seconds." These are scalars. Others also need a direction to make sense: "5 blocks northeast" or "9.8 m/s² downward." These are vectors. In diagrams, vectors are drawn as arrows: the length shows the size and the arrowhead shows the direction.

Each scalar on the left has a vector partner on the right that adds a direction.
Scalar (size only)Vector (size and direction)
Distance: 7 blocksDisplacement: 5 blocks northeast
Speed: 20 m/sVelocity: 20 m/s north
Time: 3 s(time has no direction)
Mass: 60 kgWeight: 588 N downward (a force)

Using signs for direction. When motion is along a straight line, you can pick one direction as positive and the other as negative. If "up" is positive, then a ball moving down at 5 m/s has a velocity of −5 m/s. The minus sign is not "less than nothing": it just means "the other way." Pick your positive direction at the start of every problem and stick with it.

Idea 3: Speed and velocity

Speed tells you how fast you cover ground. Velocity tells you how fast and which way. Over a whole trip, we use averages:

average speed = total distance ÷ time

average velocity = displacement ÷ time

These can be very different. A runner who does one 400 m lap in 80 s has an average speed of 400 ÷ 80 = 5 m/s, but finishes where she started, so her average velocity is 0.

The speed at one instant, like the number on a speedometer, is the instantaneous speed. Add a direction and you have the instantaneous velocity.

A car going around a circular track at a steady 20 m/s has a constant speed, but its velocity keeps changing, because its direction keeps changing. That will matter in the next idea.

Units: 1 m/s = 3.6 km/h ≈ 2.24 mph. Usain Bolt's 100 m world record of 9.58 s (Berlin, 2009) works out to an average of about 10.4 m/s, or 37.6 km/h.

A car drives clockwise around a circular track at a steady 20 meters per second. At the top, right, bottom and left of the track, a blue velocity arrow of the same length points along the track: east at the top, south on the right, west at the bottom and north on the left. The speed never changes, but the velocity keeps changing because its direction changes. 20 m/s east 20 m/ssouth 20 m/s west 20 m/snorth Same speed, changing velocity (clockwise)
A car circling a track at a steady 20 m/s. The speed never changes, but the velocity arrow points a new way at every moment, so the velocity is always changing.

Idea 4: Acceleration

Acceleration is the rate at which velocity changes:

a = (v − u) ÷ t

Here u is the starting velocity, v is the final velocity and t is the time taken. If a car goes from 0 to 27 m/s in 9 s, its velocity rises by 3 m/s every second, so a = 27 ÷ 9 = 3 m/s². The unit m/s² reads "meters per second, per second."

Because velocity has a direction, there are three ways to accelerate:

  • Speed up: acceleration points the same way you're moving.
  • Slow down: acceleration points opposite to your motion. If forward is positive, the acceleration is negative. People often call this "deceleration."
  • Turn: even at a steady speed, changing direction is acceleration. The car on the circular track is accelerating the whole time.

A motion diagram, like a strobe photo, shows this at a glance: dots at equal time intervals spread out when an object speeds up and bunch together when it slows down.

Three motion diagrams, each showing a car's position every second as a dot, numbered 0 to 4 seconds. Top: the dots are evenly spaced, so the speed is steady and the acceleration is zero. Middle: starting from rest, the gaps grow in the ratio 1 to 3 to 5 to 7, so the car is speeding up and its acceleration points forward. Bottom: the gaps shrink 7 to 5 to 3 to 1 as the car slows to a stop, so its acceleration points backward, opposite its motion. Steady speed: equal gaps, a = 001234Speeding up from rest: gaps grow 1 : 3 : 5 : 701234Slowing to a stop: gaps shrink 7 : 5 : 3 : 101234 a, forwarda, backward Numbers are seconds. Every car moves to the right.
Motion diagrams with one dot per second. Equal gaps mean steady velocity. Growing gaps mean speeding up; shrinking gaps mean slowing down. Steady acceleration from rest makes gaps in the ratio 1 : 3 : 5 : 7.

Myth: "Negative acceleration always means slowing down." Not quite. The sign only tells you the direction of the acceleration. A ball falling down while you call up positive has negative velocity and negative acceleration, and it's speeding up. An object slows down when its velocity and acceleration have opposite signs.

Idea 5: Motion graphs

Graphs let you see a whole trip at once. The two most useful ones put time on the horizontal axis.

Position–time graphs

On a position–time graph, the slope is the velocity:

velocity = slope = rise ÷ run = change in position ÷ change in time

  • A flat line: position isn't changing, so the object is at rest.
  • A straight sloping line: steady velocity. Steeper means faster. In the graph, the navy line rises 16 m in 2 s, so v = 16 ÷ 2 = 8 m/s.
  • A curve getting steeper: the velocity is increasing, so the object is accelerating.
  • A line sloping down: negative velocity, moving back toward the origin.
A position-time graph with three lines. A navy straight line rises steadily from 0 to 40 meters in 5 seconds; a slope triangle shows it climbs 16 meters every 2 seconds, a velocity of 8 meters per second. A flat gray line stays at 20 meters: an object at rest. An orange curve starts flat and gets steeper, reaching 50 meters at 5 seconds: an object speeding up. time (s)position (m) 01234501020304050 2 s16 m speeding up:slope grows navy: steady 8 m/s,constant slope at rest: slope 0
Position–time graph. Flat line: at rest. Straight line: steady velocity (here 8 m/s). Upward curve: speeding up.

Velocity–time graphs

On a velocity–time graph, two things matter:

  • The slope is the acceleration. In the graph, the velocity rises 4 m/s in 2 s, so a = 4 ÷ 2 = 2 m/s². A flat line means zero acceleration (steady velocity), not "stopped."
  • The area under the line is the displacement. Here the shaded shape is a trapezoid: ½ × (4 + 16) × 6 = 60 m. For steady velocity, the area is a rectangle: velocity × time, which is just distance = speed × time.

Area below the time axis counts as negative displacement: the object is moving backward.

A velocity-time graph. A straight line rises from 4 meters per second at time 0 to 16 meters per second at 6 seconds. A slope triangle shows the velocity rising 4 meters per second every 2 seconds: an acceleration of 2 meters per second squared. The shaded area under the line, a trapezoid, equals the displacement: one half times 4 plus 16, times 6, which is 60 meters. time (s)velocity (m/s) 0123456048121620 2 s 4 m/s slope = acceleration 4 ÷ 2 = 2 m/s² area = displacement ½ × (4 + 16) × 6 = 60 m
Velocity–time graph for an object speeding up from 4 m/s to 16 m/s in 6 s. The slope gives the acceleration, 2 m/s². The shaded area gives the displacement, 60 m.

A graph is not a picture of the path. A position–time graph that slopes up and then down doesn't mean the object went over a hill. It means the object moved away from the origin and then came back. Always read the axis labels first.

Idea 6: The constant-acceleration equations

When the acceleration stays the same the whole time, five quantities describe the motion. Many textbooks call these the suvat equations, after the letters:

  • s = displacement (m)
  • u = starting (initial) velocity (m/s)
  • v = final velocity (m/s)
  • a = acceleration (m/s²)
  • t = time (s)

Some books write v0 or vi for the starting velocity and d or Δx for displacement. The physics is the same. Each equation leaves out one of the five letters:

The five constant-acceleration equations. Use them only when the acceleration doesn't change.
EquationLeaves outWhere it comes from
v = u + atsThe definition of acceleration, rearranged
s = ½(u + v)taAverage velocity × time (the trapezoid area)
s = ut + ½at²vPut the first equation into the second
s = vt − ½at²uSame idea, worked from the end
v² = u² + 2astEliminate t from the first two

How to choose

  1. Draw a sketch and choose which direction is positive.
  2. List the three things you know and the one you want.
  3. Pick the equation that leaves out the one quantity you don't know and don't need.
  4. Substitute, solve, and check the units and the sign.

Watch the "hidden" numbers. "Starts from rest" means u = 0. "Comes to a stop" means v = 0. "Dropped" means u = 0 and a = 9.8 m/s² downward. "At the top of its flight" means the vertical velocity is 0.

Idea 7: Free fall

Drop a ball and it speeds up as it falls. Near Earth's surface, if air resistance is small enough to ignore, every object falls with the same acceleration, whatever its mass:

g = 9.8 m/s², straight down

That means a falling object gains about 9.8 m/s of downward velocity every second: 9.8 m/s after 1 s, 19.6 m/s after 2 s, 29.4 m/s after 3 s. Because it keeps speeding up, it falls farther each second than the one before: 4.9 m in the first second, but 78.4 m in total after 4 s. Free fall is just constant acceleration, so the suvat equations work with a = g.

Thrown upward? The ball slows by 9.8 m/s every second on the way up, stops for an instant at the top, then speeds up by 9.8 m/s every second on the way down. Its acceleration is 9.8 m/s² downward the whole time, even at the top, where its velocity is momentarily zero. Without air, the trip up and the trip down take the same time.

A ball dropped from rest, drawn at its position every second, to scale. After 0, 1, 2, 3 and 4 seconds it has fallen 0, 4.9, 19.6, 44.1 and 78.4 meters, and its speed is 0, 9.8, 19.6, 29.4 and 39.2 meters per second. Blue arrows beside it grow by the same amount each second. The gaps between positions grow in the ratio 1 to 3 to 5 to 7. timefallenspeed 80 m 0 s0 m0 m/s1 s4.9 m9.8 m/s2 s19.6 m19.6 m/s3 s44.1 m29.4 m/s4 s78.4 m39.2 m/s Blue arrows: velocity, +9.8 m/s every second. Air resistance ignored.
A ball dropped from rest, shown once per second and drawn to scale. Its speed grows by 9.8 m/s every second, and the distance fallen grows with the square of the time (no air resistance).

What about air resistance? In real air, a falling object pushes air out of the way, and the air pushes back. The faster it falls, the bigger that drag. Eventually drag can balance the object's weight, and it stops speeding up: it has reached its terminal velocity. A feather or a sheet of paper reaches its terminal velocity almost at once, which is why it drifts down. A skydiver falling belly-first with arms and legs spread out reaches a terminal velocity of very roughly 200 km/h (about 55 m/s), and much faster head-first. On this page, unless we say otherwise, we ignore air resistance.

The hammer and the feather

On the Moon there is no air. At the end of the last moonwalk of Apollo 15 in 1971, astronaut David Scott held up a 1.32 kg aluminum geologic hammer and a falcon feather and let them go together. As NASA's film shows, they fell side by side and hit the ground at the same moment. They fell more slowly than they would on Earth, because the Moon's gravity gives an acceleration of only about 1.62 m/s².

Idea 8: Projectile motion

A projectile is anything that's thrown or launched and then moves under gravity alone: a basketball shot, a long jumper, water from a hose. Its curved path is easier to understand than it looks, thanks to one big idea:

Horizontal and vertical motion are independent.

  • Horizontally, nothing pushes or pulls (ignoring air), so the horizontal velocity stays constant.
  • Vertically, the projectile is in free fall, with an acceleration of 9.8 m/s² downward, exactly like a dropped ball.

Roll one ball off a table while you drop another from the same height at the same instant: they hit the floor together. The rolling ball also moves sideways, but its sideways motion doesn't change how quickly it falls. Put the two motions together and the path is a curve called a parabola.

Two balls leave the same height at the same moment. A blue ball is dropped straight down; an orange ball is thrown sideways. Their positions are shown every second. Dashed horizontal lines show that at every moment both balls are at exactly the same height, falling 1, 3, 5 and 7 units in successive seconds. The orange ball also moves the same distance sideways each second, shown by equal-length arrows. Both reach the ground together. 0 s1 s2 s3 s4 s dropped thrown sideways, same moment Same height at every moment: gravity pulls both the same way. Sideways steps stay equal.
A ball dropped and a ball thrown sideways at the same moment are at the same height at every instant. The thrown ball's sideways steps stay equal because its horizontal velocity doesn't change. Spacing not to scale.

Launching at an angle

Launch at speed v and angle θ above level ground. First split the velocity into two parts using trigonometry:

across: vx = v cos θ    up: vy = v sin θ

Then treat each direction separately. For a launch from the ground that lands at the same height, with no air:

time in the air: T = 2v sin θ ÷ g

maximum height: H = (v sin θ)² ÷ 2g

range: R = v² sin 2θ ÷ g

Because sin 2θ is largest when 2θ = 90°, the longest range comes at 45°. Angles the same amount above and below 45°, like 30° and 60°, give the same range: the higher one just takes longer and flies higher.

Paths of five balls launched from the same spot at the same speed, with no air, at 15, 30, 45, 60 and 75 degrees. The 45 degree path, drawn thick in orange, lands farthest. The 30 and 60 degree paths, in blue, land at the same place. The 15 and 75 degree paths, dashed gray, also land at the same place as each other, closer in. Higher angles go higher and stay up longer. 75°60°45°30°15° 15° and 75° 30° and 60° 45° 45°: farthest 30° and 60°: same range 15° and 75°: same range Same launch speed, no air
Launches at the same speed and five different angles, with no air. 45° goes farthest. 30° and 60° land at the same spot, and so do 15° and 75°.

Real life isn't airless. Air resistance shortens the range of real balls, and for most thrown or kicked balls the best angle turns out a bit lower than 45°. Launching from above the landing level, like a shot-putter releasing from shoulder height, also lowers the best angle. The 45° rule is exact only for level ground with no air.

Worked examples

Try each one on paper first, then open the solution. Use g = 9.8 m/s² and ignore air resistance. Answers are rounded to 3 significant figures.

1. One lap

A runner completes one 400 m lap of a track in 80 s, finishing where she started. Find her average speed and her average velocity.

Show solution
Average speed = distance ÷ time = 400 ÷ 80 = 5 m/s
Displacement = 0, so average velocity = 0 ÷ 80 = 0 m/s

2. Walking to a friend's

You walk 3 blocks east and 4 blocks north in 10 minutes. Each block is 100 m. Find the distance, the displacement, your average speed and your average velocity.

Show solution
Distance = 300 + 400 = 700 m
Displacement = √(300² + 400²) = 500 m, 53° north of east
Time = 10 × 60 = 600 s
Average speed = 700 ÷ 600 ≈ 1.17 m/s
Average velocity = 500 ÷ 600 ≈ 0.833 m/s, 53° north of east

3. Merging onto the highway

A car speeds up from rest to 27 m/s in 9.0 s. What is its acceleration? How far does it travel in that time?

Show solution
a = (v − u) ÷ t = (27 − 0) ÷ 9.0 = 3.0 m/s²
s = ½(u + v)t = ½ × (0 + 27) × 9.0 = 122 m (121.5 m)

27 m/s is about 97 km/h, or 60 mph.

4. Braking distance

A car moving at 20 m/s brakes with an acceleration of −5.0 m/s². How long does it take to stop, and how far does it travel while braking?

Show solution
Time: v = u + at → 0 = 20 − 5.0t → t = 4.0 s
Distance: v² = u² + 2as → 0 = 400 − 10s → s = 40 m

Double the speed and the braking distance goes up four times, because of the v² term.

5. Reading a graph

A cyclist speeds up steadily from 2 m/s to 8 m/s in 3 s, then rides at a steady 8 m/s for 5 s. Sketch the velocity–time graph and find the acceleration in the first part and the total displacement.

Show solution
Slope of first part: a = (8 − 2) ÷ 3 = 2 m/s²
Area of the trapezoid: ½ × (2 + 8) × 3 = 15 m
Area of the rectangle: 8 × 5 = 40 m
Total displacement = 15 + 40 = 55 m

6. Dropped from a roof

A ball is dropped from a roof 20 m high. How long does it take to land, and how fast is it moving just before it hits?

Show solution
Take down as positive: u = 0, a = 9.8 m/s², s = 20 m
s = ½at² → t = √(2 × 20 ÷ 9.8) = √4.08 ≈ 2.02 s
v² = 2as = 2 × 9.8 × 20 = 392 → v ≈ 19.8 m/s

That's about 71 km/h, which is why roofs need railings.

7. Straight up

You throw a ball straight up at 14.7 m/s. How long does it take to reach the top, how high does it go, and when does it land back in your hand?

Show solution
Take up as positive: u = 14.7, a = −9.8, and v = 0 at the top
Time up: 0 = 14.7 − 9.8t → t = 1.5 s
Height: 0 = 14.7² − 2 × 9.8 × s → s = 216.09 ÷ 19.6 ≈ 11.0 m
The trip down takes as long as the trip up: lands after 3.0 s

8. Off the edge of a table

A marble rolls off a table 1.25 m high at 3.0 m/s. How long is it in the air, and how far from the table does it land?

Show solution
Vertical (starts with no vertical velocity): 1.25 = ½ × 9.8 × t² → t = √(2.5 ÷ 9.8) ≈ 0.505 s
Horizontal (steady 3.0 m/s): x = 3.0 × 0.505 ≈ 1.52 m

A marble that just slips off the edge would land at the same moment, right next to the table.

9. A soccer kick

A ball is kicked from the ground at 20 m/s, 30° above the horizontal. Find its time in the air, maximum height and range.

Show solution
vx = 20 cos 30° ≈ 17.3 m/s; vy = 20 sin 30° = 10 m/s
T = 2 × 10 ÷ 9.8 ≈ 2.04 s
H = 10² ÷ (2 × 9.8) ≈ 5.10 m
R = 20² × sin 60° ÷ 9.8 ≈ 35.3 m

Check: vx × T = 17.32 × 2.041 ≈ 35.3 m. Kicked at 60°, it would land in the same place.

Try it: the projectile launcher

Launch a ball

v = u + at calculator

Things to try

  • Keep the speed fixed and try 30°, 45° and 60°. Which goes farthest? Which two land in the same place?
  • Double the speed from 10 m/s to 20 m/s at 45°. The range doesn't double. How many times bigger is it, and why?
  • Switch to the Moon. With the same launch, how many times farther does the ball go? Compare that with 9.8 ÷ 1.62.
  • Launch at 90°. What happens to the range, and what is the time in the air for 14.7 m/s? Check it against worked example 7.
  • In the calculator, find how long a car takes to stop from 20 m/s at −5 m/s². Then try a positive acceleration for the same problem and read the message.

A short history: Galileo and the rolling ball

For nearly two thousand years, students learned the physics of Aristotle, which taught that heavier objects fall faster than lighter ones. Galileo Galilei (1564–1642), an Italian mathematician and astronomer, wanted to know exactly how falling objects speed up. The trouble was that a dropped ball falls too fast to time with the clocks of his day.

His solution was to slow gravity down. He described the experiment in his last great book, Discourses and Mathematical Demonstrations Relating to Two New Sciences, published in Leiden in 1638 by the Elzevir firm, while he was under house arrest near Florence after his trial by the Inquisition. A wooden board "about 12 cubits long, half a cubit wide, and three finger-breadths thick" had a straight groove cut along it, lined with smooth parchment. One end was raised one or two cubits, and a "hard, smooth, and very round bronze ball" was rolled down the groove.

To measure time, he used a water clock: water ran in a thin jet from a large vessel into a small glass during each run, and he weighed the water on an accurate balance. More water meant more time. Repeating the runs "a full hundred times," he found that the distances traveled were always "as the squares of the times," at every slope he tried. Distances in equal time intervals went up as the odd numbers 1, 3, 5, 7, exactly what steady acceleration predicts.

In the book's Fourth Day, he combined a steady horizontal motion with this accelerated vertical motion and showed that a projectile's path is a parabola. He proved that the longest range comes at 45°, which gunners already knew from experience, and that angles equally above and below 45° reach equally far.

An original drawing of the idea behind Galileo's inclined-plane experiment. A bronze ball rolls down a long, gently sloping grooved board. Marks along the board show where the ball is after 0, 1, 2, 3 and 4 equal time intervals; the distances covered in successive intervals are in the ratio 1 to 3 to 5 to 7, so total distances grow as the square of the time. Below, a water clock: water from a large vessel runs through a thin pipe into a small cup on a balance, and the weight of water collected measures the time. 012341357 Distance in each interval: 1, 3, 5, 7 Total from the start: 1, 4, 9, 16 = 1², 2², 3², 4² Water clock: weigh the water that flows out
An original drawing of the idea behind Galileo's inclined-plane experiment, not his exact apparatus. In equal times the ball covers distances in the ratio 1 : 3 : 5 : 7, so the totals go 1, 4, 9, 16. Not to scale.

Did Galileo drop balls from the Leaning Tower of Pisa? Maybe, maybe not. Galileo never described doing it himself. The story comes from Vincenzo Viviani, who became the elderly Galileo's assistant and wrote an admiring biography of him in 1654, long after the event was supposed to have happened. Many historians doubt it happened as told, or treat it as at most a demonstration rather than a careful experiment. The rolling-ball experiment, on the other hand, Galileo described himself in detail.

  1. c. 350 BCE Aristotle teaches that heavier bodies fall faster, a view taught for centuries.
  2. 1638 Galileo's Two New Sciences is printed in Leiden, describing the inclined plane and the parabolic path of projectiles.
  3. 1654 Viviani writes his account of Galileo's life, the source of the Leaning Tower story; it is first printed in 1717.
  4. 1687 Isaac Newton's Principia explains why things move as they do (Unit 2).
  5. 1971 On Apollo 15, David Scott drops a hammer and a feather on the airless Moon. They land together.

Test yourself

The Motion Check

Talk about it

  1. A car's speedometer shows speed, not velocity. What extra information would a "velocity-meter" need to show? Could a car have one?
  2. Can an object have zero velocity and still be accelerating? Think about a ball at the very top of its flight.
  3. A long jumper and a basketball player both launch themselves into the air. Why might neither of them jump at exactly 45°?
  4. Galileo timed his experiments by weighing water. What would you use today, and how could you check that the distances really go as the squares of the times?

Sources

  1. OpenStax, Physics (high school), Chapter 2 "Motion in One Dimension": 2.1 Relative Motion, Distance, and Displacement, 2.2 Speed and Velocity, 2.3 Position vs. Time Graphs, 2.4 Velocity vs. Time Graphs.
  2. OpenStax, Physics (high school), Chapter 3 "Acceleration": 3.1 Acceleration, 3.2 Representing Acceleration with Equations and Graphs.
  3. OpenStax, Physics (high school), Chapter 5 "Motion in Two Dimensions": 5.1 Vector Addition and Subtraction: Graphical Methods, 5.3 Projectile Motion.
  4. OpenStax, College Physics 2e: 2.7 Falling Objects, 3.4 Projectile Motion.
  5. Galileo Galilei, Dialogues Concerning Two New Sciences (1638), translated by Henry Crew and Alfonso de Salvio (1914). Third Day (the inclined plane, water clock and odd-number rule), on Wikisource: Dialogues Concerning Two New Sciences/Third Day. Fourth Day (projectiles; the 45° maximum range and Proposition VIII on equal ranges are on pp. 275–277), in a scan of the 1914 Macmillan edition at the Internet Archive: Two New Sciences, Fourth Day (p. 244 onward).
  6. Library of Congress catalog record for the first edition, Discorsi e dimostrazioni matematiche, intorno à due nuoue scienze (Leida: Elsevirii, 1638). https://www.loc.gov/item/55003623/
  7. Wikipedia, "Two New Sciences" (publication by Lodewijk Elzevir in Leiden; summary of the Fourth Day). https://en.wikipedia.org/wiki/Two_New_Sciences
  8. Museo Galileo, "Racconto istorico della vita di Galileo by Vincenzo Viviani" (written 1654, first printed 1717). https://catalogue.museogalileo.it/indepth/RaccontoIstoricoVitaGalileoByViviani.html
  9. Philip Ball, "Tall tales," Nature news (2005), on doubts about the Leaning Tower story. https://www.nature.com/news/2005/050613/full/news050613-10.html
  10. OpenStax, College Physics 2e, 5.2 Drag Forces (terminal velocity of a skydiver).
  11. NASA Space Science Data Coordinated Archive, "The Apollo 15 Hammer-Feather Drop." https://nssdc.gsfc.nasa.gov/planetary/lunar/apollo_15_feather_drop.html
  12. NASA, "Moon Fact Sheet" (surface gravity 1.62 m/s²). https://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html

Credits: all diagrams on this page, including the drawing of Galileo's inclined plane and the share image, are original SVG made for Jugami. The free-fall, projectile and range diagrams are drawn from the equations on this page.