Hub
Kinematics
One map: from s = v·t through acceleration and throws to energy. Each node has one short note and a next link.
Start with part 1 of the series: uniform motion. Full guide to all eight parts: Physics without mysteries.
On this page: s is distance, v is speed, t is time, a is acceleration. g is gravitational acceleration, about 9.81 m/s². v₀ is the speed at the start.
The idea
You ride a bike at steady speed, then accelerate up a short hill, then brake before a junction. To calculate that, you first name what is constant: speed, acceleration or gravity. This map helps you pick the right starting point.
Kinematics describes motion: position, velocity, acceleration, distance and time. It does not yet ask where the force comes from. First you can say what happens; then dynamics and energy say why.
From scratch, start with the simplest model: constant speed. Then:
s = v·t
The s(t) graph is a straight line, and any two of v, s, t give the third. This step builds the unit habit and the distance-versus-displacement habit in one-dimensional problems. Start with the article and the uniform-motion calculator.
A micro-check: v = 15 m/s for t = 40 s gives s = 600 m. That is a calm bike or a slow city car. If you type 54 km/h as though it were m/s, the “distance” blows up, because 54 km/h is exactly 15 m/s.
The second level is constant acceleration. Speed changes linearly: vk = v0 + a·t. Distance gains the ½·a·t² term. On the site you compute bare a = Δv/t on one page and distance at known a on another. Those are two steps in the same calculation, not two different topics.
Example: from rest to 20 m/s in 8 s gives a = 2.5 m/s². With v0 = 0, distance is s = ½·a·t² = 0.5 · 2.5 · 64 = 80 m. Someone who multiplies final v by t gets 160 m and doubles the result.
The third level is gravity as a special constant a: a = g downward, usually g = 9.81 m/s². Free fall, vertical throw, horizontal throw from a height and angled throw are variants of the same assumption: no air drag, g = const.
The table below is not a list of every formula. It is a map: which row to open when the problem says “constant speed”, “speeds up”, “brakes”, “drops from a roof” or “at angle α”. Each row has a short note and a next link.
Calculators stay split on purpose. The v-s-t tool is not the same as “distance only”. Trip average speed (stops included) is not the speedometer reading. Vertical, horizontal and angled throws have separate pages because questions about H_max, Z and flight time are often separate too.
A common beginner mistake: use s = v·t when v is not constant. If a car accelerates from rest, distance is not a hand-waved “average v times t”, but the result from the constant-a formula. Name the model first, then pick the formula.
A second common mistake: mix average and instantaneous speed. “I was doing ninety” is often a momentary reading. If the question is about the whole trip, use v̄ = s/t with total distance and total time. A trip of 90000 m in 1.5 h (5400 s) is an average 16.7 m/s, about 60 km/h, even if the meter briefly showed 90 km/h.
A third mistake: in throws, mix H_max with range Z. Maximum height and range are different quantities and often different calculators. The three-throw comparison keeps them in one table: vertical, horizontal and angled.
The bridge to dynamics is simple: when you know mass and net force, a = F/m. That a then plugs back into kinematics. For m = 1000 kg and F = 2500 N you get a = 2.5 m/s², then you can return to distance at constant a.
The bridge to energy: Ek = ½·m·v² and Ep = m·g·h can close height or speed without an explicit time, if friction is not eating energy. At v = 10 m/s the equivalent height ½v²/g ≈ 5.1 m matches the order of H_max in a vertical throw.
The inclined plane joins both bridges. Without friction a = g·sin α; with friction μ enters. Incline kinematics is again constant a, only computed differently. The incline article in Physics without mysteries walks that step.
Keep units consistent. Convert km/h to m/s (divide by 3.6), minutes to seconds, centimeters to meters. Example: 72 km/h is 20 m/s. Most “weird” results are mixed units, not bad physics.Keep units consistent. Convert mph to ft/s, minutes to seconds, inches to feet. Example: 72 km/h is 20 m/s in SI (US mode shows the ft/s counterpart). Most “weird” results are mixed units, not bad physics.
Working assumptions for this map: segments with constant v or constant a, g = const, no drag in the throws on this path, straight-line motion or planar projectile motion. When a problem adds wind, a varying force or reaction time, the map points to another row (for example stopping distance instead of bare deceleration).
Practical ritual: (1) read what is constant, (2) name the model, (3) list data in matching units, (4) pick the table row and calculator, (5) check order of magnitude against intuition (for example 50 km/h and a few seconds of braking).
The “acceleration” node is often confused with the whole accelerated-motion story. Bare a = Δv/t answers how fast velocity changes. Only when you add v0 and t (or s) do you return to distance. The map splits those questions because problem text often splits them too.
Force and energy sit at the end of the map on purpose. You may open them earlier if that matches how you learn, but kinematics without them is complete as a description. F = m·a says where a comes from. Ek and Ep say how to balance without an explicit t.
If you want a systematic path, follow Physics without mysteries: eight parts with an example and calculator fill. This map is a navigation shortcut: one question, one starting row. If after a minute you see that v is not constant, move one row down.
When stuck, return to “what is constant?”. Constant v, constant a or constant g picks a map branch more reliably than hunting a formula by wording alone. After walking the table you should point, for a typical problem, to one starting row and one calculator.
Keep three bridges in mind: constant v → s = v·t; constant a → distance with ½at²; constant g → throws and falls. Force and energy do not replace those bridges; they feed them. Once you know which bridge you are on, picking a calculator becomes boring in the good way.
How to read the table
The “Node” column is the idea, “About” is the model sketch, “Where next” is an article or calculator. Start from the row that matches the problem text, not from the first row by habit.
If a problem mixes traffic, braking and constant-v driving again, split the path into segments. Each segment gets its own map row. Do not force one formula onto everything.
What we assume
Kinematics on this map works on segments with constant v or constant a. In throws: constant g, no air drag. Force enters only as a source of a (F = m·a) or as a bridge to energy. We work in meters, seconds and m/s.Kinematics on this map works on segments with constant v or constant a. In throws: constant g, no air drag. Force enters only as a source of a (F = m·a) or as a bridge to energy. We work in feet, seconds and ft/s.
When this map is not enough
The map does not cover curved motion beyond planar projectile motion, relativity, or speed-dependent drag. When those effects matter, the formulas in the rows below are only a sketch. Then you need a model beyond constant v or constant a.
Which formula to use
| Node | About | Where next |
|---|---|---|
| Uniform motion | Constant v; solve any two of v, s, t. | Article 1 + uniform-motion calculator |
| Distance s = v·t | Distance only from known v and t. | uniform-motion-distance |
| Average speed | v = s/t for the whole trip (stops included). | average-speed |
| Acceleration | Just a = Δv/t before distance enters. | acceleration |
| Falls / throws | g = const; vertical, horizontal, angled. | Throw comparison + articles 4-6 |
| Incline | a = g sin α (no friction) and μ variant. | Article 7 + incline-no-friction |
| Force F = m·a | Where a comes from: the second law of motion. | force-f-ma |
| Ek / Ep | ½mv² and mgh: energy in motion. | Article 8 + energy calcs |
Frequently asked questions
Which node should I open if I have never computed motion?
Uniform motion first: s = v·t and the trio v, s, t. Then acceleration as a = Δv/t, then distance at constant a. Physics without mysteries follows the same order in eight parts with examples.
When is s = v·t allowed?
When speed is constant on the whole segment you compute, or when you knowingly use the constant-v model. If v rises or falls, you need constant-a formulas (or an average defined correctly through total s and t).
How does average speed differ from instantaneous speed?
Instantaneous is the “now” reading (speedometer, radar). Average is total distance divided by total time, stops included. Do not drop the average into the constant-v model without thinking.
Does kinematics require F = ma?
Not to describe motion when a is given. F = m·a is a bridge: force and mass give a, then you return to kinematics. On the map that bridge is a separate row so description and cause stay distinct.
How do I choose between horizontal and angled throw?
If the initial velocity is only horizontal from height h, use horizontal throw. If it is at angle α to the horizontal, use angled throw. H_max, Z and t side by side are on the throw comparison page.
Why is there a separate “distance only” calculator?
Because many problems ask only for s = v·t from known v and t. The full v-s-t tool solves any two of the three fields. Splitting reduces wrong-field clicks and teaches which data you actually have.
Where is braking on the map?
It is constant deceleration: kinematics like acceleration, but v falls to zero. The series has a dedicated braking part, and the tools include decelerated motion plus stopping distance with reaction time.
Does energy belong to kinematics?
Strictly it is already mechanics with a balance. On the map it sits at the end because Ek and Ep often close height or speed without computing time. Treat it as a bridge, not the first row.
Which value of g should I use?
Usually 9.81 m/s², or 10 m/s² if the problem says so. Consistency matters more than a magic constant: the same g in the whole solution and in the calculator.
What if a problem mixes traffic, braking and constant-v driving again?
Split the path into segments. Each segment gets its own model: trip average separately, braking separately, constant v separately. The map exists so you do not force one formula onto everything.