Example 1
A short playground slide, modeled with no friction: 30° and a length of 2.4 m. From rest the ride takes under a second.
a = 4.90 m/s², t = 0.99 s, v = 4.85 m/s
Inclined Plane Calculator
Enter the angle α in degrees and, if you want, the ramp length L and the acceleration g (an empty g field means 9.81 m/s²). The calculator finds the acceleration along a smooth incline from a = g sin α. With a length, and a start from rest, it also reports the slide time and the final speed. On a 30° ramp that is 5 m long, the acceleration is 4.90 m/s², the slide takes 1.43 s, and the speed at the bottom is 7.00 m/s. The model does not subtract friction μ and does not take a nonzero initial speed.
Result
Enter the angle α. Length L and g are optional. Acceleration, the components, and — once you enter a length — the slide time appear here.
For motion on an incline with kinetic friction, use the incline with friction calculator; for the friction force T = μ·N alone, use the friction force calculator.
How it works
g9.81 m/s²
sin 30°0.50
a4.90 m/s²
t over 5 m1.43 s
A frictionless incline is a board tilted at an angle α to the horizontal, down which a body slides with no drag. The acceleration along the board is a = g sin α, where g is the gravitational acceleration. That is the component of weight parallel to the ramp, written with Newton’s second law as a = F/m, and used earlier by Galileo to slow free fall enough to time it. General-physics textbooks state it the same way (Halliday, Resnick, and Walker; Young and Freedman).
The calculator uses the numbers you enter: the angle α in degrees, an optional length L in meters, and an optional g. An empty g field means 9.81 m/s². At 30° the sine is 1/2, so 9.81 × 0.5 = 4.905 m/s², and the panel, at two decimal places, shows 4.90 m/s². Over 5 m, from rest, the time is t = √(2L/a) = 1.43 s, the final speed is v = √(2aL) = 7.00 m/s, and the height drop is h = L sin α = 2.50 m.
What the result panel shows. Besides the acceleration, the calculator reports sin α, cos α, the perpendicular component g⊥ = g cos α (at 30° the panel shows 8.50 m/s²), the share a/g as a percent (at 30° that is 50.00%), and, once you enter L, the time, the final speed, and the height drop. The diagram under the result labels those two components. The chart compares acceleration at 10°, 20°, 30°, 45°, 60°, and 75° for the g you entered.
Limits of the model. The formula assumes μ = 0, constant g, and a start from rest when you ask for time and speed. The calculator does not subtract friction: that is the incline-with-friction calculator, a = g(sin α − μ cos α). It does not ask for an initial speed; motion with a nonzero v₀ is covered by the uniformly accelerated motion calculator. Air resistance is omitted, so on a long ramp the result overstates the speed.
The unit switch in the header changes length to feet and acceleration to ft/s². The angle stays in degrees. Figures in the text stay in SI units as the reference. A comma and a period in the fields are the same decimal separator. The angle must lie from 0° to 90°: at 0° the acceleration is 0.00 m/s², and at 90° it equals g.
The examples run from a 2.4 m slide at 30° (0.99 s), through a 1.2 m board at 15° (0.97 s), to the main case of 5 m at 30° (1.43 s and 7.00 m/s), and to the Moon at g = 1.62 m/s² (3.51 s). The button “Fill this example” enters those same numbers and refreshes the panel, the diagram, and the chart.
Formula
a = g sin(α)
From rest over a length L: t = √(2L/a), v = √(2a L), h = L sin(α). Perpendicular component: g⊥ = g cos(α).
Model: resolution of weight on a frictionless incline (Stevin 1586 for the component; Galileo, Discorsi 1638, for the motion; in Newton’s notation a = g sin α). Friction μ is zero here. Order of computation: sine of the angle in degrees, product with g, and, when L is entered, the square roots of 2L/a and of 2aL.
Theory, derivation, and domain
Definition of a frictionless incline
An inclined plane is a surface tilted at an angle α to the horizontal. With no friction, the only driving force along the board is the component of weight parallel to the ramp. That component is mg sin α, so the acceleration, independent of mass, is a = g sin α. The sine grows from 0 on a level board to 1 on a vertical board, and the acceleration grows the same way: from zero to the full g.
The length L is measured along the board, not horizontally. The height the body loses is h = L sin α. At 30° and 5 m that is half the length, 2.50 m. The potential energy tied to that height becomes kinetic energy, which gives the same final speed as v = √(2aL), because aL = g h.
History: Stevin, Galileo, and a slowed fall
Simon Stevin, in De Beghinselen der Weeghconst (1586), showed that the component of weight along an incline compares with the full weight as the height of the incline compares with its length, which is sin α. His chain-of-spheres experiment (the clootcrans) was about equilibrium, not about timing a slide. The component he established is the one this calculator uses.
Galileo Galilei, in the Discorsi of 1638, rolled balls down an incline because a vertical drop was too fast for a water clock. He found that the distances covered in successive equal times stand as 1 : 3 : 5 : 7, so distance grows with the square of time. The incline reduced the acceleration by the factor sin α and made the motion measurable. The formula a = g sin α is the later Newtonian writing of that same relation.
Deriving a = g sin α from Newton’s second law
Step 1. Point one axis down the ramp and the other straight out from the board. The weight mg splits into mg sin α along the incline and mg cos α perpendicular to it. Step 2. With μ = 0 the friction force is zero, and the normal force cancels the perpendicular component. The net force along the board is therefore just mg sin α.
Step 3. Newton’s second law, F = ma, gives ma = mg sin α. Mass cancels, which leaves a = g sin α. Step 4. At α = 30° and g = 9.81 m/s² the product is 4.905 m/s². The panel shows 4.90 m/s². The perpendicular component g cos 30° is about 8.496 m/s², and the panel shows 8.50 m/s². The share a/g = sin α = 50.00%.
Slide time and final speed from rest
When the acceleration is constant and the initial speed is zero, distance along the board grows as L = ½ a t². Inverting that gives t = √(2L/a). The speed at the bottom is v = a t, which for this time is the same as v = √(2aL). At 30° and 5 m: a = 4.905 m/s², t = 1.43 s, v = 7.00 m/s.
An empty L field leaves only a. An entered L = 0 gives a time of 0.00 s and a speed of 0.00 m/s, because there is no distance to cover. At α = 0 and L > 0 the acceleration is zero, so the calculator does not divide by zero: it reports that there is no slide.
Assumptions: zero friction, constant g, no drag
The formula a = g sin α is exact under three assumptions. First, friction is zero, both static and kinetic. Even a small μ reduces the acceleration by g μ cos α; once μ is at least tan α, a block at rest does not start. Second, g is constant along the whole length. For ramps a few meters or a few tens of meters long, that is an excellent approximation.
The third assumption is about the time and the speed, not about the acceleration: the initial speed is zero, and air resistance is omitted. On a short, smooth slide the error is small. On a 40 m ski-jump ramp at 35° the model gives 21.22 m/s, while air and ski friction lower the real speed. Mass does not return to the formula while friction and drag stay out.
The angles 0° and 90°, and the friction boundary
At 0° the board is level, sin α = 0, and a = 0. At 90° the board is vertical, sin α = 1, and a = g: that is free fall, which a separate fall-time calculator also computes. Between those ends the acceleration rises smoothly. The chart on this page shows six angles at the g you enter.
Once the board does resist the slide, this calculator no longer describes the motion. The incline-with-friction calculator subtracts μ cos α inside the parentheses. The friction force T = μ N, with no incline angle, is covered by the friction-force calculator. Keep the three results apart: here μ is zero from the start.
Sources
The definition, the derivation, and the history come from textbooks and historical sources: S. Stevin, De Beghinselen der Weeghconst, Leiden 1586; G. Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze, Leiden 1638; D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, Newton’s laws and the inclined plane; H. D. Young and R. A. Freedman, University Physics, dynamics and the incline; R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. I, Chapter 12 (forces and components); BIPM, the SI Brochure. This page is for educational purposes.
How to use
- Enter the angle α in degrees, for example 30 for half of g or 15 for a slower slide.
- Leave the length L empty if you want acceleration only, or enter 5 when you want time and speed over five meters.
- Leave g empty for Earth (9.81 m/s²). For the Moon, enter 1.62.
- Click Calculate or wait for the live result. At 30° and 5 m the calculator shows 4.90 m/s², 1.43 s, and 7.00 m/s.
- Check sin α, cos α, the perpendicular component, the share of g, and the diagram. With friction, open the incline with μ. The Clear button removes the entries and hides the panel.
Symbol glossary
a is the acceleration along the incline, α the angle, L the board length, and g the gravitational acceleration. At 30° and 5 m the panel shows 4.90 m/s², 1.43 s, and 7.00 m/s.
- a
- Acceleration along the incline [m/s²], a = g sin α. At 30° the panel shows 4.90.
- α
- Incline angle to the horizontal [°]. Range 0 to 90. Not radians.
- L
- Length along the board [m]. An empty field leaves a only.
- g
- Gravitational acceleration [m/s²]. Empty means 9.81. The Moon is 1.62.
- sin α
- The fraction of g that remains along the board. At 30° the panel shows 0.50, which is 50.00% of g.
- g⊥
- Perpendicular component g cos α [m/s²]. At 30° the panel shows 8.50. The normal force cancels it.
- h
- Height drop [m], h = L sin α. At 30° and 5 m it is 2.50 m.
- t, v
- Slide time [s] and final speed [m/s] from rest. At 30° and 5 m: 1.43 s and 7.00 m/s.
- μ
- Friction coefficient. On this page it is zero. A nonzero value belongs on the incline with friction.
Real-life examples
Example 2
A lab board raised to 15° over 1.2 m. The small angle slows a marble enough that a stopwatch can time the run.
a = 2.54 m/s², t = 0.97 s, v = 2.47 m/s
Example 3
A carton slides down a board leaned on a loading dock: 20° and 1.5 m. The acceleration is about one third of g.
a = 3.36 m/s², t = 0.95 s, v = 3.17 m/s
Example 4
A smooth loading ramp at 8° and 12 m. The acceleration is 13.92% of g, so the slide is slow.
a = 1.37 m/s², t = 4.19 s, v = 5.72 m/s
Example 5
Snow on a 40° roof, 6 m along the slope, computed with no friction. Real snow drags, so this result is an upper bound on the acceleration.
a = 6.31 m/s², t = 1.38 s, v = 8.70 m/s
Example 6
A classroom model of a ski-jump ramp: 35° and 40 m. Over that length, air resistance already cuts the real speed below 21.22 m/s.
a = 5.63 m/s², t = 3.77 s, v = 21.22 m/s
Example 7
The same 30° and the same 5 m, on the Moon at g = 1.62 m/s². The slide takes more than twice as long as on Earth.
a = 0.81 m/s², t = 3.51 s, v = 2.85 m/s
Example 8
A steep attic board: 60° and 3 m. Sin 60° is √3/2, so the panel acceleration is 8.50 m/s².
a = 8.50 m/s², t = 0.84 s, v = 7.14 m/s
Example 9
A near-vertical chute at 80° and 3 m. The motion is close to free fall: the acceleration is 9.66 m/s².
a = 9.66 m/s², t = 0.79 s, v = 7.61 m/s
Example 10
The worked example on this page: a 30° incline, 5 m long, with Earth g. The height drop is 2.50 m, the time is 1.43 s, and the speed at the bottom is 7.00 m/s.
a = 4.90 m/s², t = 1.43 s, v = 7.00 m/s
Common questions
What does this calculator compute, and from which formula?
The calculator finds the acceleration of a body on a frictionless incline from a = g sin α. The angle α is in degrees, and an empty g field means 9.81 m/s². At 30° the panel shows 4.90 m/s², because 9.81 × sin 30° is 4.905 m/s² and two decimal places round that product to 4.90.
Which units should I enter for the angle, the length, and g?
Enter the angle α in degrees, the length L in meters, and g in meters per second squared. After you switch the header to US units, L is in feet and accelerations are shown in ft/s². A comma and a period are the same decimal separator.
What happens if I leave g or L empty?
An empty g field means 9.81 m/s². An empty L field means the calculator reports acceleration only, without slide time or final speed. An entered length adds t = √(2L/a) and v = √(2aL), for a start from rest.
Why does the acceleration not depend on mass?
The force along the incline is mg sin α. Dividing by mass in Newton’s second law cancels the mass and leaves a = g sin α. A heavier block and a lighter block slide with the same acceleration as long as friction is zero.
Do I enter the angle in degrees or in radians?
In degrees. Entered 30 means thirty degrees, not 30 radians. The calculator converts degrees to radians before it evaluates the sine. The angle has to lie from 0° to 90°.
When do slide time and final speed appear?
Only after you enter a length L and the angle gives a > 0. At 30° and 5 m the time is 1.43 s, the final speed is 7.00 m/s, and the height drop h = L sin α is 2.50 m. Both motion formulas assume a start from rest.
What do 0° and 90° mean?
At 0° the sine is zero, so the acceleration along the board is 0.00 m/s² and a block at rest does not start. At 90° the sine is 1, the acceleration equals g, and the motion is vertical free fall. At 90° and 5 m the time is 1.01 s.
Where are friction and air resistance?
Kinetic friction is covered by the separate incline-with-friction calculator, which uses a = g(sin α − μ cos α). This calculator sets μ = 0. Air resistance is omitted too, so on a long, steep ramp a real slide is slower than the result.
Does the time formula work if the body is already moving?
No. The time t = √(2L/a) and the speed v = √(2aL) are for a start from rest. If the body already has a speed, use the uniformly accelerated motion calculator. The acceleration a = g sin α itself stays valid at a constant angle and zero friction.
How do I calculate a slide on the Moon?
Enter g = 1.62 m/s². At 30° and 5 m the acceleration is 0.81 m/s², the time is 3.51 s, and the final speed is 2.85 m/s. The height the body loses is h = L sin α, and at the same angle and length it stays 2.50 m.
Why is there no result when the angle is out of range?
The school model of an incline uses an angle from 0° to 90°. A negative angle, or one past 90°, does not describe a slide down this board. While the angle is out of range, or the length is negative, the calculator shows a message instead of a result.
How do I calculate acceleration on a frictionless incline?
Use a = g·sin α. At α = 30° and g = 9.81 m/s² you get 4.905 m/s², shown as 4.90 m/s² in the panel.
How long is a 5 m slide at 30° from rest?
About 1.43 s. First a = 4.90 m/s², then t = √(2L/a). Speed at the bottom is √(2aL) ≈ 7.00 m/s.
How is this different from an incline with friction?
Here μ = 0 and a = g·sin α. With kinetic friction a = g(sin α − μ·cos α) on the incline-with-friction page.
Why does mass not enter a = g sin α?
The weight component along the ramp is m·g·sin α, and a = F/m, so m cancels. With no friction or drag, a heavy and a light block slide the same way.
Does the calculator handle a nonzero initial speed?
No. Time and final speed assume a start from rest. Motion with a known v₀ belongs on the uniformly accelerated motion calculator.
Sources
The definition, the history, and the derivation come from established physics sources.
- Wikipedia — Inclined plane
- Wikipedia — Inclined plane
- Wikipedia — Simon Stevin
- Wikipedia — Two New Sciences (Galileo, 1638)
- Feynman Lectures on Physics, Vol. I, Chapter 12 — Characteristics of Force
- BIPM — The International System of Units (SI)
- NIST — SI units
- Stevin, S., De Beghinselen der Weeghconst, Leiden 1586.
- Galilei, G., Discorsi e dimostrazioni matematiche intorno a due nuove scienze, Leiden 1638.
- Halliday, D.; Resnick, R.; Walker, J., Fundamentals of Physics — Newton’s laws, the inclined plane.
- Young, H. D.; Freedman, R. A., University Physics — dynamics, inclined plane.
Page updated November 3, 2026.