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A free fall hill shape gives a rider a weightless sensation. To give this weightless sensation over a hill, the hill is designed to have the same shape as the path of a ball being thrown off the top of a hill. Shape is determined by how fast the roller coaster car travels over the hill. The faster the coaster travels over the hill the wider the hill must be. There are two ways to apply projectile motion concepts to design the hill's shape. The first way is to calculate the coaster's position as if it drove off a cliff.
The position equation is as follows.

This can be rewritten as



Catching the Rider
There comes a certain point on the free-fall drop where the track needs to redirect the riders. Otherwise the riders will just plummet into the ground. This point is the transition point from free-fall to controlled acceleration. This point is also the maximum angle of a hill. This angle can be in virtually any range from 35° to 55°.

For the bottom section of the track, the new equation has the desired outcome of changing the direction of the coaster from a downward motion to a purely horizontal motion. The track will need to apply a vertical component of velocity to reduce the coaster's vertical velocity to zero. The track will also need to increase the horizontal velocity of the coaster to the value determined from energy relationships. The velocity at the bottom of the hill is determined from

which is

This simplifies to

where vB is the horizontal velocity at the bottom of the hill. The value for vB will be used in later calculations.

Recall one of the original horizontal equations.

x = xo + (vxo)t +(1/2)(ax)t2


substituting in our expression for "t" yields,

where vxo is the horizontal velocity of the coaster at the transition angle and vyo is the vertical component of the velocity at the transition angle. "ay" is calculated from


Where "vy"is the final vertical velocity of zero, "vyo" is the vertical component of the velocity at the transition point, and "y" is the distance left to fall from the transition point to the ground. The horizontal velocity is determined from a parameter decided upon by the engineer. The engineer will want to limit the g forces experienced by the rider. This value will be the net g's felt by the rider. These net g's are the net acceleration.

these values are plugged back into the original equation and x values are calculated as a function of y.


(It's the curved hill in front.)

If you use or find this page useful or have any comments, please contact the author so maybe he'll do more. Author: Tony Wayne


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