Magnus EffectBeta
Fly a spinning ball beside a spinless one and see how far it bends
This tool is a beta version still being stabilized. Please let us know if a result looks off.
A pitcher throws to a plate 18.44 m away. Aiming at the same spot, backspin makes the ball drop less and arrive high (a four-seam fastball); topspin makes it dive (a curveball). Thrown as aimed with no spin, the ball lands in the dirt before the plate — the faded ball shows exactly that, so rescue it with spin.
Goal Pass through the strike zone (6 tries)
Ball
| With spin | No spin | |
|---|---|---|
| Distance flown | – | – |
| Peak height | – | – |
- Spin factor S
- 0.00
- Magnus acceleration
- 0.0 m/s²
Conditions
Previous tries
Why does a spinning ball curve?
A spinning ball drags the air around with its surface. On one side the surface moves with the airflow and speeds it up; on the other it moves against the flow and slows it down. Pressure is lower where the air is faster, so the ball is pushed that way. That push is the Magnus force.
With backspin the air over the top is sped up and the ball is lifted. Part of gravity is cancelled, so the ball seems to float and carries farther. With topspin the ball is pressed down instead and dives. A curling free kick is the same force aimed sideways, and a baseball curve or slider differs only in the tilt of the spin axis.
The strength follows the spin factor S = ωr/v. Faster spin and bigger balls mean more force, but past S ≈ 0.3 the lift coefficient saturates — piling on spin stops helping. A golf ball's dimples hold the boundary layer on, cutting drag while boosting Magnus lift, so the same swing flies much farther.
None of this happens without air. Turn the air off in the conditions panel and no amount of spin bends the path — it stays a plain parabola. The Magnus force comes from the ball interacting with air, not from the spin by itself.
F = ½ρ C_L A v²S = ωr / v (spin factor)C_L ≈ 1.5 S, saturating past S ≈ 0.23
Working…
0.0
It hasn't frozen — the result will appear when the calculation finishes.
About Magnus Effect
This tool treats the ball as a point and integrates its equations of motion with fourth-order Runge–Kutta in steps of half a millisecond or less. Three forces act: gravity; drag ½ρC_D·A·v², opposite to the ball's velocity relative to the air; and the Magnus force ½ρC_L·A·v², pointing along the velocity rotated 90° toward the spin. The airflow around the ball is not simulated; the forces come from coefficients of the kind measured in wind tunnels.
Everything happens in one vertical plane. The spin axis points out of the screen, so backspin lifts the ball and topspin pulls it down. A real curveball breaks sideways as well as down; here you only see the vertical part.
Why a spinning ball gets pushed
On a ball flying with backspin, the top surface moves with the oncoming air and the bottom surface against it. The boundary layer clings longer on top and leaves the ball further round toward the bottom, so the wake is deflected downward. Pushing air down means the ball is pushed up. That reaction is the Magnus force.
Ideal-flow theory describes the same thing with circulation Γ. Adding a flow that circles a rotating cylinder to a uniform wind speeds up one side and slows the other, and the lift per unit length is given by the Kutta–Joukowski theorem, L′ = ρ·U·Γ. That formula ignores viscosity and separation, so for a real ball it predicts far more force than you get.
The lift model inside the tool
The Magnus force depends on the spin factor S = ωr/v, the ratio of surface speed from rotation to flight speed. The tool uses C_L = k × min(0.35, 1.5·S): proportional at small S, capped at 0.35 above S ≈ 0.23. The factor k reflects the surface: 1.0 for baseball and soccer ball, 1.1 for tennis, 1.2 for ping-pong and 1.3 for the dimpled golf ball.
Alan Nathan (2008) fired baseballs from a pitching machine at 50–110 mph with 1,500–4,500 rpm of spin (S from 0.09 to 0.6) and found that at fixed S the lift coefficient does not depend strongly on speed. The rise-then-saturate shape in this tool is a simplification of that trend, not a fitted curve for each ball. Spin decays exponentially with a per-ball time constant (20 s for a baseball, 4 s for a ping-pong ball).
Switch the air off
In the free lab, turning the air off removes drag and the Magnus force together. However much spin you add, the spinning and non-spinning balls follow the same path. The Magnus force is not a property of the ball but an interaction between ball and air. In vacuum the range matches v²·sin2θ/g exactly.
Ships, aircraft and balls
The same effect can drive a ship. In the 1920s Anton Flettner fitted the schooner Buckau with two tall cylinders spun by electric motors in place of sails, and such devices are still called Flettner rotors. In 2008 the wind-turbine maker Enercon launched the cargo ship E-Ship 1 with four rotors 25 m tall and 4 m across. Wind crossing a spinning cylinder produces a Magnus force at right angles to the wind, and the ship uses it as thrust.
Worked examples
Curveball challenge: 30 rev/s on a baseball
- Input
v = 38 m/s, topspin 30 rev/s, r = 3.66 cm, m = 145 g- Output
S ≈ 0.18, C_L ≈ 0.27, Magnus acceleration ≈ 7.0 m/s² (about 71% of g)
S = 2π×30×0.0366/38 = 0.182 and C_L = 1.5×0.182 = 0.272. a = ½×1.225×π×0.0366²×0.272×38²/0.145 ≈ 6.99. Over the roughly 0.5 s to home plate that adds about ½×7×0.5² ≈ 0.87 m of extra drop.
Free kick: 10 rev/s on a soccer ball
- Input
v = 27 m/s, 10 rev/s, r = 11 cm, m = 430 g- Output
S ≈ 0.26, so C_L hits the 0.35 cap; acceleration ≈ 13.8 m/s²
The large radius makes S big even with modest spin. The acceleration exceeds gravity, so with backspin the ball floats almost straight for a while.
Golf drive: 50 rev/s backspin
- Input
v = 66 m/s, 50 rev/s (3,000 rpm), r = 2.13 cm, m = 45.9 g- Output
S ≈ 0.10, C_L ≈ 0.20, acceleration ≈ 16 m/s²
At launch the Magnus force is 1.7 times gravity. As the ball slows the force falls with v², which is why the end of the flight drops steeply.
How to use it
- Pick a scenario and set speed, launch angle and spin with the sliders or by dragging the scene.
- After the throw, compare with the faded no-spin ball to see what the spin changed.
- Adjust the spin while watching the spin factor and the Magnus-as-percent-of-gravity readout.
Frequently asked questions
Can I see a curveball break sideways?
No. The tool computes a vertical plane only. A real curveball with a tilted spin axis breaks down and sideways at once; it is the same force pointed in a different direction.
Why doesn't more spin always mean more curve?
The tool's lift coefficient saturates at 0.35 above S ≈ 0.23. Real measurements also show C_L rising more slowly at high S.
Why is the drag coefficient fixed?
For simplicity each ball uses one value (0.35 for a baseball, 0.24 for a golf ball and so on). Real C_D varies with Reynolds number and spin, and for a smooth sphere it falls sharply around the drag crisis. Golf-ball dimples bring that fall down to lower speeds.
Would Kutta–Joukowski be more accurate?
That result is for a cylinder in inviscid flow and greatly overestimates the force on a real ball. Measured lift coefficients are the better choice.
Sources
- A. M. Nathan, The effect of spin on the flight of a baseball, Am. J. Phys. 76, 119–124 (2008)
- NASA Glenn Research Center — Ideal Lift of a Spinning Ball (Kutta–Joukowski)
- HyperPhysics — Kutta-Joukowski Lift Theorem
- NASA Glenn Research Center — Drag of a Sphere
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