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Physics-based simulation of a ball on a roller coaster (or a marble on a track) where the ball is also attached to a spring.

You can change the track shape by clicking a button below. You can drag the spring anchor point or drag the ball to change the starting position. Try modifying gravity, damping or spring stiffness.

The math behind the simulation is shown below. Also available: source code, documentation and how to customize.

Buckel

Schleife

Kreis

Lemniscate

Oval

Spirale

Physics of the Roller Coaster with Spring

This simulation is a variation of the Simple Roller Coaster, please see that page for more information on the physics and how the simulation is done. The only difference here is that the spring force is added to the differential equation.

roller coaster forces

Define the following variables:

(p_x, p_y) = position of ball in 2 dimensional space
(q_x, q_y) = fixed end of spring
k = slope of track at point P
R = rest length of spring
c = stiffness of spring (constant)
θ = angle between the spring and tangent vector
m = mass
g = gravity constant

The width and height of the spring is given by:

s_x = q_x − p_x
s_y = q_y − p_y

The length of the spring is then √(s_x² + s_y²) . The magnitude of the spring force is c (√(s_x² + s_y²) − R) . The direction of the spring force is towards vec(Q), the fixed end of the spring. The component of the spring force that is parallel to the track is

F = cos(θ) c (√(s_x² + s_y²) − R)

(1)

It is this parallel-to-the-track component of the spring force that will accelerate the ball along the track. Consider a couple of cases to convince yourself of this: If the spring is perpendicular to the track, then θ ≈ 90° and cos(θ) ≈ 0 so the spring doesn't accelerate the ball. Conversely, if θ ≈ 0 then the spring will accelerate the ball in the positive track direction. Or, if θ ≈ 180° then the spring accelerates the ball in the negative track direction.

We can find cos(θ) by using the formula for the angle between two vectors A, B

cos θ =	A · B
	\|A\| \|B\|

where the numerator is the dot product, and the denominator is the product of the magnitudes. Define our two vectors by

A = (1, k) the slope vector, pointing in the direction of increasing p
B = (s_x, s_y) a vector in the direction of the spring force

This leads to

cos θ =	s_x + k s_y
	√(1 + k²) √(s_x² + s_y²)

Put this expression into equation (1) and combine with the gravity and friction forces developed for the Simple Roller Coaster to get

a =	−g k	− ^b⁄_m v +	c (√(s_x² + s_y²) − R) (s_x + k s_y)
	√(1 + k²)		m √(1 + k²) √(s_x² + s_y²)

(2)

where b = damping constant and v = velocity. Keep in mind that the slope k and spring stretch s_x, s_y are functions of the position p .

Besides this change to the force equation, the simulation is similar to the Simple Roller Coaster. Please see that page for more information on how the simulation is implemented.

This web page was first published June 2002.

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