**PROJECTILE MOTION AND ROLLER COASTER
HILLS**

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

**EXAMPLE CALCULATIONS**

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°.

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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 v_{B} is the horizontal
velocity at the bottom of the hill. The value for v_{B} will be used in later calculations.

Recall one of the original horizontal equations.

x = x_{o} + (v_{xo})t +(^{1}/2)(a_{x})t^{2}

substituting in our expression for "t" yields,

where v_{xo} is the horizontal
velocity of the coaster at the transition angle and v_{yo} is the vertical component of the velocity
at the transition angle. "a_{y}" is calculated from

Where "v_{y}"is the final vertical
velocity of zero, "v_{yo}" 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

"ROLLER COASTER PHYSICS" TABLE
OF CONTENTS ... PHYSICS PAVILION TABLE OF
CONTENTS

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