Arrows that carry a size and a direction. Once you can picture them, velocity, acceleration and force all start to make sense. Drag everything. Break nothing.
Keep this color code in mind — it's used everywhere below.
Suppose I tell you the temperature is 25°C. That's the whole story — one number. But if I say "there's a wind of 30 km/h," your next question is "which way?" A number alone isn't enough; wind has a direction.
That little difference splits all of physics into two families:
A single number (with a unit). No direction.
A number plus a "which way."
Notice the pairs: distance vs displacement, speed vs velocity. The first of each is a scalar (just "how much"), the second is a vector ("how much, and which way"). Walk 5 km in a circle and your distance is 5 km but your displacement is zero — you ended up where you started.
A vector is anything with a magnitude (a size) and a direction. We draw it as an arrow: the length is the size, and where it points is the direction.
Let's make those two words precise, because the whole tutorial rests on them.
A quantity that is fully described by a single number together with a unit — nothing else is needed. "The bag has a mass of 5 kg" or "the trip took 10 s" tells you everything. Scalars can be compared, added, and multiplied just like ordinary numbers: 3 kg + 2 kg = 5 kg, no questions asked.
A quantity that needs two pieces of information to be complete: a magnitude (how much) and a direction (which way). "A force of 10 N" is unfinished — 10 N in what direction? Until you say "10 N to the right," you haven't named the vector. Because of that extra direction, vectors do not add like plain numbers (we'll see exactly how they add in Steps 3–4).
The "size" or "amount" of a vector, always a non-negative number with a unit — for a velocity it's the speed, for a force it's how hard the push is. We write the magnitude of a vector v as |v| or just plain v. The direction is the other half of the story and is usually given as an angle.
A quick word on notation: in textbooks a vector is printed in bold (v) or with a tiny arrow over it (v⃗), while its magnitude is written in ordinary italic (v). On this page we lean on the color code and arrows instead of heavy symbols, but it's worth recognizing the notation when you meet it. Also remember: every physical quantity carries a unit (metres, seconds, newtons…); a bare number without a unit is meaningless in physics.
Grab the orange dot below and drag it around. You're building a vector with your hand. A longer arrow means a bigger magnitude; turning it changes the direction. That's the entire object.
An important habit: a vector doesn't care where it sits. A 5 N push to the right is the same vector whether it acts in Delhi or on the Moon. Only its length and direction define it.
Every vector-arrow has a tail (where it starts) and a head (the pointed end). The length from tail to head represents the magnitude — drawn to scale, so an arrow twice as long means twice the magnitude. The way the arrow points is the direction.
We pin down the direction with an angle, measured from the positive x-axis (the rightward horizontal), turning counter-clockwise for positive angles. So 0° points right, 90° points straight up, 180° points left, and 270° (or −90°) points down. Magnitude + angle together name any vector in a plane.
Two vectors are equal when they have the same magnitude and the same direction — even if they're drawn in different places on the page. Reverse a vector's direction (turn it 180°) while keeping its length and you get its negative, written −v: same size, opposite way. "5 m/s east" and "5 m/s west" are negatives of each other.
Here's the trick that makes vectors easy to actually calculate with. Instead of dealing with a slanted arrow, we replace it with two arrows at right angles: one purely horizontal (the x-part) and one purely vertical (the y-part).
Think of giving directions in a city with a grid of streets. You could say "walk 500 m at 53° north of east" — or you could just say "3 blocks east" and "4 blocks north." Same destination, but the second version is way easier to follow. Breaking a vector into x and y is exactly this.
The orange arrow, its x-part, and its y-part form a right triangle. The orange arrow is the hypotenuse (length V), at angle θ from the horizontal. From basic trig:
x-part = V·cos θ y-part = V·sin θ
And to go backwards, from the parts to the whole, it's just Pythagoras: V = √(x² + y²). That's all the math the rest of this page needs.
A component of a vector is how much of it points along one chosen axis. The x-component is the part along the horizontal axis; the y-component is the part along the vertical axis. Each component is itself just a number (with a sign and unit) — it tells you "this much of the vector goes this way."
Resolving means splitting a single slanted vector into its perpendicular components. For a vector of magnitude V at angle θ above the horizontal:
Vₓ = V·cos θ V_y = V·sin θ
Going the other way (components → whole vector) uses Pythagoras for the magnitude and the arctangent for the angle: V = √(Vₓ² + V_y²) and θ = tan⁻¹(V_y / Vₓ).
Components carry a sign. If the vector points left, its x-component is negative; if it points down, its y-component is negative. The sign is the direction along that axis — so once you've resolved a vector, each axis becomes a simple "positive or negative number line," which is exactly why components make problems easy.
A ball is kicked at 10 m/s at 30° above the ground. Resolve its velocity:
vₓ = 10·cos 30° = 10 × 0.87 ≈ 8.7 m/s (forward)
v_y = 10·sin 30° = 10 × 0.50 = 5.0 m/s (upward)
Check with Pythagoras: √(8.7² + 5.0²) = √(75.7 + 25) ≈ √100.7 ≈ 10 m/s. ✓ The two components really do rebuild the original 10 m/s.
Any slanted vector = one horizontal arrow + one vertical arrow. This is called "resolving" the vector.
This is the single most useful — and most surprising — fact about vectors. The horizontal and vertical directions are independent. What happens sideways has zero effect on what happens up-and-down, and vice-versa.
Classic experiment: take two identical coins. Drop one straight off a table. At the exact same moment, flick the other one sideways so it flies off horizontally. Which lands first?
They land at the same instant. The flicked coin travels much farther across the room, but gravity pulls both downward in exactly the same way. The sideways motion didn't make it fall any slower or faster. Press the button and watch:
Why does this work? Gravity is a force that points straight down — it only changes the vertical part of the motion. There is nothing pushing the coins sideways once they leave your hand, so the horizontal part just coasts along unchanged. The two stories run in parallel and never interfere.
Motion (or a force) along one axis has no effect on motion along a perpendicular axis. The horizontal "story" and the vertical "story" of an object unfold completely separately and share only one thing: the same clock. A push purely sideways changes only the sideways motion; a pull purely downward changes only the downward motion.
This is more than a fun fact — it's the working method of 2-D physics. Whenever a problem looks complicated because things move diagonally, you split every vector into x and y, solve two ordinary 1-D problems, and then recombine the answers at the end. Two easy problems instead of one hard one.
Why "same clock" matters: the only link between the horizontal and vertical stories is time. In the coin demo, both coins fall for the same number of seconds, so they hit the floor together — but during those seconds the thrown coin also travels sideways, which the dropped coin doesn't. Same vertical story, different horizontal story, one shared timer.
Treat x and y as two separate 1-D problems. That's why components are so powerful.
Now let's attach meaning to the arrows. Velocity is a vector that tells you how your position is changing: its length is your speed, and it points in the direction you're moving.
If something moves diagonally, it has both a sideways speed vₓ and an up/down speed v_y at the same time. Set the two parts, then press Run. Watch the object's horizontal progress and vertical progress grow into a right triangle — the slanted path is just those two added together.
So "moving in 2D" is really just "having an x-velocity and a y-velocity at the same time." The familiar x and y on a graph are simply the two directions we split everything into.
Position is where an object is, given as coordinates (x, y) measured from some agreed origin. Displacement is the vector from a starting position to an ending position — a straight arrow "from here to there." It's different from distance (a scalar): displacement cares only about start and end points, not the path taken.
A vector equal to the rate of change of position — how much displacement happens each second, and in which direction. Its unit is metres per second (m/s). Its x-component vₓ is how fast the position's x changes; its y-component v_y is how fast y changes.
Speed is just the magnitude of velocity — the bare number on the speedometer, with no direction attached: speed = |v| = √(vₓ² + v_y²). A car going round a roundabout at a steady 50 km/h has constant speed but changing velocity, because its direction keeps changing.
Average vs instantaneous: average velocity is total displacement ÷ total time over a whole trip; instantaneous velocity is the velocity at one exact moment (what the arrow shows at any instant in the animation). When we say "velocity" without qualification in this tutorial, we mean the instantaneous kind.
Velocity tells you how position changes. Acceleration tells you how velocity changes. It's a "change of a change." And here's the part most students miss: any change in velocity is an acceleration — getting faster, getting slower, or turning.
Watch the three cases below. The teal arrow is velocity (where you're going); the amber arrow is acceleration (how the velocity is changing right now).
a along v → speeding up.
a opposite v → slowing down (braking).
a sideways to v → turning. In a steady circle the speed never changes, yet the object is always accelerating, because its direction keeps changing. The amber arrow points to the center — that's "centripetal" acceleration, and it's why you feel pushed outward on a merry-go-round.
Velocity = where you head. Acceleration = how that heading is being changed.
A vector equal to the rate of change of velocity — how much the velocity vector changes each second, and in which direction. Its unit is metres per second per second (m/s²). Crucially, it responds to any change in the velocity vector, whether that's a change in length (speed) or a change in where it points (direction).
Because velocity is a vector, there are exactly three ways it can change — and each one is an acceleration:
The special case of pure turning. When an object moves in a circle at constant speed, its acceleration points straight toward the centre of the circle (the word "centripetal" means "centre-seeking"). The speed is constant, yet the object is accelerating every instant because its direction is always changing.
Students often think "if it's moving, something must be pushing it." Not true. The real chain of cause and effect goes like this:
A force is a push or a pull. When there's a net force on an object, it produces an acceleration in the same direction, following Newton's famous rule:
F = m · a ⟺ a = F / m
The acceleration then gradually changes the velocity, and the velocity moves the position. So force comes first in the causal chain — but notice it doesn't directly set the velocity. That's why a moving object with no force keeps gliding at constant velocity forever (Newton's first law): nothing is there to change it.
A push or a pull on an object — an interaction that, if unopposed, changes the object's motion. Force is a vector (it has a direction), and its unit is the newton (N). Examples: gravity pulling down, a hand pushing a door, friction dragging backward, a magnet attracting iron.
Real objects often feel several forces at once. The net force is the single vector you get by adding all of them together (tip-to-tail, as vectors). Only the net force matters for motion: if the forces cancel to zero, the object behaves as if no force acts at all.
Mass is a scalar, measured in kilograms (kg), that measures how much an object resists being accelerated — its inertia. The same force gives a small mass a big acceleration and a large mass a small one, because a = F/m.
First law (inertia): with zero net force, an object keeps doing what it was doing — at rest stays at rest, and moving keeps moving at constant velocity in a straight line. Motion needs no force to continue, only to change.
Second law: net force equals mass times acceleration, F = m·a. The acceleration points in the same direction as the net force, and is bigger for bigger forces and smaller for bigger masses.
The key subtlety: force sets the acceleration, not the velocity. So a force can point in a completely different direction from the motion — and when it does, it gradually bends the velocity rather than instantly redirecting it. That's exactly what scenario ② below shows.
Let's watch the chain happen. Pick a situation and press Run. The force (violet) stays constant; keep your eye on the velocity (teal) — every moment, the force nudges it a little, and the faint teal arrows left behind show how the velocity is changing over time.
Near Earth, gravity pulls every object with the same downward acceleration, about g = 9.8 m/s². (Weight is just the force version: F = m·g downward.) Because this force is purely vertical, it only touches the y-part of the motion — exactly the independence idea from Step 4.
Launch a ball below. Watch the truth of it: the horizontal speed stays constant the whole flight (no sideways force), while the vertical speed steadily shrinks, stops at the top, then grows downward. Add those two parts at every instant and you get the smooth curved path — a parabola.
Near Earth's surface every freely-falling object speeds up downward at the same rate, g ≈ 9.8 m/s², regardless of its mass (ignoring air resistance). A heavy stone and a light pebble dropped together land together — a direct consequence of a = F/m, since a heavier object feels a proportionally bigger gravitational pull.
Mass (kg) is how much matter is in an object — the same everywhere in the universe. Weight is the force of gravity on that mass, W = m·g, measured in newtons and pointing downward. On the Moon your mass is unchanged but your weight is about one-sixth, because the Moon's g is smaller.
A projectile is any object moving under gravity alone after launch. Its motion is the sum of two simple pieces: constant-velocity horizontal motion (no sideways force) and constant-acceleration vertical motion (gravity). Combine a steady horizontal advance with an evenly-changing vertical position and the resulting curve is a parabola — the smooth arch you see in a thrown ball, a water fountain, or a basketball shot.
This is the classic textbook setup. An electron is fired horizontally into the gap between two parallel charged plates. Between the plates there is a uniform electric field E (drawn as the faint vertical lines) pointing from the + plate to the − plate. The field pushes the charge with a constant force F = qE — so, exactly like gravity on a thrown ball, the path is a parabola.
Because that force is purely vertical, the horizontal speed vₓ never changes; only the vertical speed v_y grows. Launch it below, and flip between an electron (−) and a positive charge (+) to see the deflection reverse.
A basic property of matter that comes in two kinds, positive and negative, measured in coulombs (C). An electron carries a tiny negative charge. Charges of the same sign repel; opposite signs attract.
A region of space where a charge feels a force. The electric field is a vector defined as the force per unit positive charge, with unit newtons per coulomb (N/C). A uniform field has the same strength and direction everywhere — like the field in the gap between two parallel charged plates — which is why it produces a constant force, just like gravity.
A charge q sitting in a field E feels a force F = qE. For a positive charge the force points along the field; for a negative charge (like the electron) it points opposite the field. Either way the force is constant, giving a constant acceleration a = qE/m and therefore a parabolic path — identical maths to a projectile under gravity.
It's nothing more than "resolve into x and y, and the two are independent." Say the plates have length L and the charge enters with horizontal speed vₓ.
tan θ = v_y / vₓ = qEL / (m·vₓ²)
A heavier charge (bigger m) or a faster entry (bigger vₓ) bends less; a stronger field or longer plates bends it more. Flipping the sign of the charge flips the direction of v_y, so the beam deflects the opposite way.
You just used one set of ideas — vectors, components, and F = m·a — to describe a thrown ball and an electron. The arrows don't care whether the push comes from gravity or an electric field; the math is identical.
| Quantity | What it is | Vector? |
|---|---|---|
| position (x, y) | where the object is | yes |
| velocity | how fast position changes + direction | yes |
| acceleration | how velocity changes (faster/slower/turning) | yes |
| force | the push/pull that causes acceleration | yes |
| speed, mass, time | just sizes, no direction | no (scalars) |
Three habits to carry away:
| Term | Meaning | Vector? | Unit |
|---|---|---|---|
| Scalar | quantity with size only | no | varies |
| Vector | quantity with size and direction | yes | varies |
| Magnitude | the "size" of a vector, |v| | no | varies |
| Direction | which way a vector points (an angle) | — | degrees |
| Component | part of a vector along one axis (Vₓ, V_y) | no (signed) | varies |
| Resolving | splitting a vector into perpendicular parts | — | — |
| Resultant | the single vector that several parts add up to | yes | varies |
| Position | where an object is, (x, y) | yes | m |
| Distance | length of the path travelled | no | m |
| Displacement | straight vector from start to end | yes | m |
| Speed | magnitude of velocity | no | m/s |
| Velocity | rate of change of position | yes | m/s |
| Acceleration | rate of change of velocity | yes | m/s² |
| Centripetal accel. | acceleration toward the centre in circular motion | yes | m/s² |
| Force | a push or pull; cause of acceleration | yes | N |
| Net force | vector sum of all forces on an object | yes | N |
| Mass | amount of matter; resistance to acceleration | no | kg |
| Weight | force of gravity on a mass, m·g | yes | N |
| g | acceleration due to gravity (≈9.8) | yes | m/s² |
| Projectile | object moving under gravity alone | — | — |
| Parabola | the curved path of a projectile | — | — |
| Electric charge | property causing electric force (q) | no | C |
| Electric field | force per unit positive charge (E) | yes | N/C |