Roller Coaster Design

Hills and Dips
(Projectile Motion, Potential Energy and Kinetic Energy)

HILLS AND DIPS

One of the most basic parts of a ride is going from the top of a hill to the bottom. There are two basics ways designs transport riders to the bottom of a hill. The first is called the “Speed Run.”

SPEED RUN DROPS

A speed run is designed to give the rider the feeling of accelerating faster and faster without the feeling of weightlessness. It simulates being in a powerful car with the accelerator held down to the floor. It is a straight piece of track that connects a high point to a low point.
The increase in velocity of the car comes from lost gravitational potential energy being converted into kinetic energy. Next to a horizontal straight piece, the speed run is the easiest piece of the track to design and analyze.

When coasting up to a new height the calculations are the same as the example shown above. The shape of the hill does not matter. See the “Intro to Design” section, step 7, for an example of these up hill calculations.

FREE FALL DROPS
One of the biggest thrills on a roller coaster is the free fall as a rider travels over a hill. The easiest way to experience free fall is to hang from a tall height and drop to the ground. As a person falls he experiences weightlessness. As long as a person travels in the air like a projectile he will feel weightless.
Suppose a ball traveled off a table, horizontally, at 10 ^m/s. The ball’s path would look like the path shown below.

Now suppose the ball traveled off the table top on a shallow angled ramp. It would look like the one below.

A straight, “speed run,” drop does not match the fall of a rider over a hill.

To give riding more of a thrill, the designer needs to design the shape of the hill to match the falling ball.

The only problem with curve above is the impact with the floor. To alleviate this problem another curve scoops the balls as they descend. This makes the ride smooth and survivable for the rider.

The speed at the bottom of a free fall drop is calculated the same way as the speed at the bottom of a speed run drop. The only difference is the shape of the hill from the top to the bottom.

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

x in meters from hill's center	h in meters from the hill's top
0.00	0
4.52	1
6.39	2
7.82	3
9.04	4
10.10	5
11.07	6
11.95	7
12.78	8
13.55	9
14.29	10

These two tables use the second equation to calculate the position. For example, the calculation for line two in the v_o=10 m/s table looks like the equation below,

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°.

To calculate the angle of the hill use the same methods you would use to calculate the impact velocity of a projectile hitting the ground. In other words use the vetical and horizontal velocities to calculate the angle at that position.

Example:
Calculate the freefall’s angle for a coaster then is traveling over the top at exactly 10 m/s, when the rider has dropped 4 meters from the crest of the hill.

Solution
To solve this the horizontal and vetical velocities at thei pointin space need to be calculated.
In the absence of any HORIZONTAL forces. the horizontal velocity will not change from the initial velocity. It is still 10 m/s.
The vertical velocity is found from kinematics.

- - - - - - - - - - - -

After dropping going over the top of the hill at 10 m/s nad dropping down 4 meters, the angle of the free fall shaped track is 41.5° beneath the horizontal. This kind of process lends itself well to a spreadsheet calculation.

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)t²

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.)

by Tony Wayne ...(If you are a teacher, please feel free to use these resources in your teaching.)

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