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Acceleration Down an Incline with Friction

a=g(sin⁡θ−μkcos⁡θ)a = g\left(\sin\theta - \mu_k \cos\theta\right)
sig figs

Worked example: 30° ramp with μk 0.2 → 3.2048 m/s². Press Try an example to run it live, then adjust anything.

Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value. Change any unit and the answer follows.

Constant used: built into this formula, no need to enter
gn=9.80665 m/s2g_{n} = 9.80665\ \text{m/s}^{2}Standard acceleration of gravity · exact

See also

Acceleration Down a Frictionless Incline: Both use down slope acceleration and incline angle.Normal Force on an Incline (N = mg cos θ): Both use incline angle.Weight Component Along an Incline (mg sin θ): Both use incline angle.Kinetic Friction Force (f = μₖN): Both use coefficient of kinetic friction.
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Acceleration Down an Incline with Friction explained

aμkθ

Gravity pulls the block down-slope with mg sin θ while friction drags back with μₖ mg cos θ; divide the difference by m and the mass drops out, leaving a = g(sin θ − μₖ cos θ). A 30° ramp with μₖ = 0.2 gives 9.80665 × (0.5 − 0.2 × 0.866) ≈ 3.20 m/s², a third slower than the frictionless 4.90 m/s². If the bracket comes out negative, the block was never sliding in the first place: the slope sits below the angle of repose, and the honest answer is a = 0.

Solving for θ uses the amplitude-phase identity sin θ − μ cos θ = √(1+μ²)·sin(θ − arctan μ), which is why the rearranged angle carries an arctan and an arcsin. The equation is the working model behind ski-slope grooming, luge run design and conveyor chute angles, all of which are chosen to land the acceleration in a narrow, controllable band.

Acceleration Down an Incline with Friction formula

a=g(sin⁡θ−μkcos⁡θ)a = g\left(\sin\theta - \mu_k \cos\theta\right)
Where
  • aa= Acceleration (m/s²)
  • θ\theta= Incline angle (°)
  • μk\mu_k= Coefficient of kinetic friction
Revised 31 October 2026
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formula.expert. (2026, October 31). Acceleration Down an Incline with Friction Calculator. https://formula.expert/solvers/incline-acceleration-with-friction

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