Ask someone how fast a car is going and they will tell you its speed. Ask whether it is speeding up and you have moved to a different physical quantity entirely. Confusing speed, velocity and acceleration is the most common early physics error, and it makes everything afterwards feel harder than it is.
Speed, velocity, and what direction adds
Speed is distance divided by time — a magnitude, always positive:
Average speed = total distance ÷ total time
Velocity is displacement divided by time — a vector, so it has direction and can be negative:
Average velocity = displacement ÷ time
The two are identical for motion in a straight line with no reversal, and different the moment you turn around. Drive 10 km forward and 10 km back: you have travelled 20 km (speed 20 km over the trip) but ended where you started, so displacement is 0 and average velocity is 0. The speedometer was busy the whole time; the velocity was not.
Worked example: a train covers 120 km in 1.5 hours. Average speed = 120 ÷ 1.5 = 80 km/h. The velocity calculator returns speed, displacement and the distinction side by side for exactly this reason.
Acceleration is the change, not the state
Acceleration is the rate of change of velocity. For constant acceleration over a time interval t:
a = (v₁ − v₀) ÷ t · d = (v₀ + v₁) ÷ 2 × t
Displacement uses the mean velocity because under constant acceleration velocity changes linearly with time — the area under the velocity–time graph is a triangle plus a rectangle, averaging out to (v₀ + v₁)/2. The third standard form, v² = v₀² + 2ad, removes time entirely and is what you use when you know a velocity and a distance but not the duration.
Worked example: a car accelerating
A car goes from rest to 25 m/s in 10 s: a = (25 − 0) ÷ 10 = 2.5 m/s², and it covers d = (0 + 25)/2 × 10 = 125 m.
Quoted in road terms, 0–100 km/h (27.8 m/s) in 9 s is about 3.1 m/s² — which is why that figure is the one manufacturers advertise. "Acceleration in units of 0–60 mph" is the same number with different clothing on.
A starting from rest with a negative acceleration is the braking case: sign alone tells you whether the object is speeding up, slowing down, or reversing.
Constant velocity means zero acceleration
This trips people up more than any other point. An object moving smoothly along a straight line at 20 m/s has a = 0, not a. Acceleration describes a change; there is nothing to change about a constant velocity.
What is often confused with acceleration is force. Newton says force equals mass times acceleration, so zero acceleration means the net force is zero — but the forces themselves may be large and cancelling. A book on a table has a weight pulling down and the table pushing up, and they balance exactly. A satellite, by contrast, is in free fall: gravity is the only force, and it accelerates constantly while the velocity vector changes direction every instant.
What falling actually does
Near Earth's surface, gravity gives about 9.81 m/s² downward (often rounded to 9.8 or 10). A falling object gains 9.81 m/s of speed every second: 9.81 after one, 19.6 after two, 98.1 after ten, having fallen roughly 490 m in those ten seconds.
Heavier objects do not fall faster. The gravitational force on a heavy object is larger (F = mg), but its inertia is proportionally larger too, and the mass cancels in F = ma. This is why a feather and a coin hit the floor together on the Moon, where there is no air to fight them.
In air, the picture changes: drag rises with speed until it balances the weight, and the object approaches a terminal velocity — a skydiver, a thrown ball and a raindrop all do this, and it is why a skydiver stops accelerating.
Sign conventions and the three cases
Choose a positive direction, then read the sign off directly:
- a > 0 — velocity increasing in the positive direction: speeding up (or turning, if direction changes).
- a < 0 — velocity decreasing: slowing down, or accelerating the other way.
- a = 0 — constant velocity: straight line, unchanging speed.
The case that generates the most mistakes is turning. A car holding 20 m/s around a bend has zero acceleration in the scalar sense but a large one as a vector: its velocity direction is changing. That is centripetal acceleration, a = v²/r, pointing at the centre of the corner. At 20 m/s through a 50 m radius bend, that is 8 m/s² — nearly as hard on the passengers as braking at similar g.
Choosing which kinematic equation
There are four standard forms, and they are not interchangeable — picking the wrong one is the usual cause of a wrong answer:
- v = v₀ + at — velocity, acceleration, time. Use when acceleration is constant and you want a velocity.
- d = v₀t + ½at² — displacement from the start, when you do not know the final velocity.
- v² = v₀² + 2ad — velocities and distance, with time eliminated. Use in projectile problems and kinematics questions that never mention time.
- d = (v₀ + v)t/2 — average velocity form, best when the velocities at the two ends are known.
The practical test: identify which quantity is missing from the problem, then choose the form that does not contain it. A problem giving a, t and v₀ wants the first; one giving a, d and v₀ wants the third. Notice that two of the four are direct definitions and two come from combining them — which is why there are four and not more.
Free fall: the special case worth memorising
Dropping something with v₀ = 0 and a = −9.81 m/s² makes the arithmetic clean, and it is worth learning in this form because it is a very common exam and practical case:
Distance: d = ½gt² · Speed: v = gt · Impact speed from height h: v = √(2gh)
That last one is the most useful: it tells you the speed at which something hits the ground from a known height, with no time involved. A 1.5 m fall gives v = √(2 × 9.81 × 1.5) ≈ 5.4 m/s; a 30 m fall gives ≈ 24.3 m/s. Note again that mass is absent — every object from a dropped coin to a dropped anvil lands at the same speed in a vacuum, which is the fact that made Galileo's contemporaries object to the idea and that the second law explains.
Motion graphs: reading them without confusion
Three graph types carry the same information, and confusing them is a very common source of wrong answers.
Position–time: the slope is velocity. A horizontal line means at rest; a straight line with constant slope means constant velocity; a curve means changing velocity, and the curvature is the acceleration.
Velocity–time: the slope is acceleration, and the area under the curve is displacement. A line above zero with the x-axis as baseline gives positive displacement; a curve crossing the axis means it reverses direction, and the negative area subtracts.
Acceleration–time: the area gives the change in velocity.
The area rule is the one worth memorising, because "displacement is the area under a velocity–time graph" answers a question the formulas cannot when acceleration is not constant. On that graph, distance is the area, and no further calculation is needed.
Frequently asked questions
What is the difference between speed and velocity?
Speed is distance divided by time and is always a positive magnitude. Velocity is displacement divided by time and includes direction, so it can be zero or negative even when you have been moving.
Why is acceleration measured in m/s²?
It is the rate of change of velocity with time — metres per second, gained or lost, every second. A reading of 2.5 m/s² means the velocity is 2.5 m/s higher now than it was one second ago.
Does constant velocity mean zero acceleration?
Yes. Constant velocity means the acceleration is zero. It does not mean no forces act — an object moving at constant velocity in a straight line has balanced forces, while a satellite in orbit has essentially none.
Why does everything fall at the same rate regardless of mass?
In a vacuum, gravity gives every object the same acceleration g ≈ 9.81 m/s² because the gravitational force grows with mass (F = mg) in exactly the proportion that inertia also grows (F = ma), so mass cancels. Air resistance is what makes a feather and a coin behave differently.