Coaster Dynamics: Physics Primer Chapter 12



Coaster Dynamics
Physics Primer

Chapter 12

Conservation of Energy

Having established the concepts of work, and kinetic and potential energy, we are now in a position to study one of the most important unifying principles in all sciences... the conservation of energy.

Conservation of Energy for Conservative Work

As was discussed in the previous chapter, when a ball is thrown up into the air, its kinetic energy is converted to gravitational potential energy... and then re-converted back to kinetic energy when it falls back downward. This is an example of energy conservation with a special type of work:   conservative work.

Work is conservative if it can be performed without any loss of energy. And, the forces that produce conservative work are called conservative forces.

In the case of the ball being thrown upward, the kinetic energy of its initial motion is converted to gravitational potential energy without loss. Similarly, the potential energy is re-converted to kinetic energy without loss when it falls back downward. The fundamental principle in action is that the total amount of conservative energy is constant, and that the energy merely switches from one form to another. Stated mathematically,

    (12-1)         E = KE + PE = 0

where E is the change in total energy of the object, KE is the change in kinetic energy, and PE is the change in potential energy.

Despite its deceptive simplicity, the principle of energy conservation is an extremely powerful tool, which can be employed to analyze many interesting problems... yielding useful conclusions without the need to know many specific details of the system (which would otherwise necessitate formidable calculations). This is demonstrated in the two examples at the end of this chapter.

General Principle of Energy Conservation

Kinetic and potential energy are only two types of energy that may exist... and they both happen to be conservative. However, there are many other forms of conservative energy, such as the energy in a metal spring, hydraulic potential energy, electrical capacitance, etc. In addition, there are many forms of nonconservative energy and forces, such as friction and electrical resistance.

For nonconservative work, energy is "lost." In actuality, the energy is not truly lost, it is merely converted into a form that we choose to no longer include in our system for analysis. The nonconservative work done by friction and air resistance (as they apply to roller coasters) will be discussed in the next chapter.

If one views the total energy of an object or system to consist of the sum of all conservative and nonconservative energy, then the idea of energy conservation can be expanded into the:

    General Principle of Energy Conservation

Energy may be transformed from one kind to another, but it cannot be created or destroyed.

In other words, the energy of an object or system can be converted to and from one form to another... but the total energy and work associated with the energy transformations (both conservative and nonconservative) is constant. Stated mathematically,

(12-2)         E = Wc + Wnc = 0

where E is the change in total energy, Wc is the total conservative work, and Wnc is the total nonconservative work.

The General Principle of Energy Conservation is a fundamental axiom unifying many fields of physical science (eg., physics, electronics, hydraulics, chemistry, etc.). Indeed, as Albert Einstein proved with his famous equation, E = mc², even mass itself is merely another form of energy.

Motion of a Sliding Block on a Non-Linear Hill

In this analysis, we will compare the velocity of a block sliding down a frictionless non-linear hill to a block sliding down a frictionless inclined plane, as illustrated in Figure 12-A.

Figure 12-A.   Motion of a Sliding Block Down a Frictionless Non-Linear Hill and an Inclined Plane.

In both cases, we assume the block starts from rest (ie., x0 = y0 = 0), and we choose a reference frame such that vy0 = 0.

For the case of the non-linear hill, we simply note that the change in potential energy is equal to:

(12-3)         PE = mgh = mg(y - y0) = mgy.

From equation (12-1), we know that the change in kinetic energy is equal to the negative change in potential energy:

(12-4)         KE = (1/2)(m)(v²) = -mgy.

Solving equation (12-4) for v yields:

(12-5)         v² = -2gy.

Note that in equation (12-5), the value of y varies from 0 to -y (since the block is moving downward).

This result shows that the velocity of the block is a function only of the change in height, y. This is true even though the acceleration is not constant, and no matter what shape the hill has. Hence, we can conclude that for the inclined plane, the velocity is also given by equation (12-5).

No additional analysis is needed to determine the velocity on the inclined plane. However, let's derive an alternate solution for this case, just to confirm the validity of equation (12-5).

In equation (8-7) in chapter 8, the acceleration, ay, for a frictionless inclined plane was found to equal:

(12-6)         ay = (g)( cos²(ß) - 1 ).

For a constant incline angle, ß, the force and acceleration will also be constant. In this case, from equations (5-22) and (5-23) in Chapter 5, the velocity and displacement are equal to:

(12-7)         vy = vy0 + (ay)(t) = (ay)(t)

(12-8)         y = y0 + (vy0)(t) + (1/2)(ay)(t²) = (1/2)(ay)(t²).

By squaring both sides of equation (12-7), we obtain:

(12-9)         vy² = (ay²)(t²).

Solving for t² in equation (12-8), and inserting it into equation (12-9) then yields:

(12-10)         vy² = (ay²)( 2y/ay ) = (ay)(2y).

Inserting equation (12-6) into (12-10) now yields:

(12-11)         vy² = (ay)(2y) = (g)(cos²(ß) - 1 )(2y) = (2gy)(cos²(ß) - 1 ).

From the geometry in the figure, it can be shown that for motion along the surface of the plane, the velocity, v, is equal to:

(12-12)         v = (vy)/(sin(ß)).

By combining equations (12-11) and (12-12), we now get

(12-13)         vy² = ( (v)(sin(ß)) )² = (v²)(sin²(ß)) = (2gy)(cos²(ß) - 1).

The math wizards among us will now recall the trigonometric identity:

(12-14)         sin²(ß) + cos²(ß) = 1.

Using equation (12-14) to simplify (12-13), we now finally obtain:

(12-15)         v² = -2gy,

which is identical to equation (12-5).

So... we have demonstrated in this case how the principle of conservation of energy allows us to easily calculate the velocity. Alternately, even in this relatively simple situation (a constant inclined plane), it is much more difficult to do so using other means.

Motion of a Roller Coaster Train of Cars Over a Hill

An age-old debate among roller coaster enthusiasts is, what position in a roller coaster train of cars provides the most thrilling ride?

Clearly, the two most frequent answers are either the front car, or the very last car. Based on the lines for most roller coaster rides, the consensus seems to favor the front car. But... are there any real differences in the motion dynamics which might make one position more thrilling than the other?

Perhaps surprisingly, we can use the principle of conservation of energy to analyze this question, and derive some meaningful conclusions -- without ever doing any actual calculations.

As a basis for our analysis, we will investigate the speed and acceleration as you crest a hill -- when traveling over the same portion of the hilltop.

In the previous chapter, it was explained that the motion of an assembly of masses (like a roller coaster train) can be analyzed by means of its net center of gravity. It is assumed that a roller coaster train includes rigid linkages between each car -- thus, the entire train can be considered a "rigid body." That is, the total length of the train is constant (even though the shape changes with the profile of the track). Furthermore, every car in the train must move at the same speed.

Figure 12-B illustrates a roller coaster train as it goes over the top of a hill.

Figure 12-B.   Motion of a Roller Coaster Train for a Front-Car Ride.

For our analysis, we will focus our attention on the section of track between points A and C. In this first case, we consider a train whose front car moves from point A to C. In the three diagrams, the train is at position A (front car at point A), at position B (train at the top of the hill with its center of gravity at its highest point), and at position C (front car at point C). As indicated in the figure, the position of the center of gravity is lowest in position A, highest in position B, and somewhere in between in position C.

Figure 12-C illustrates a train as it goes over the hill as the last car moves from point A to C.

Figure 12-C.   Motion of a Roller Coaster Train for a Back-Car Ride.

Analogous to Figure 12-B, the three diagrams show the train at position A (last car at point A), at position B (at the top of the hill), and at position C (last car at point C). In this case, as indicated in the figure, the position of the center of gravity is lowest in position C, highest in position B, and somewhere in between in position A.

By means of the principle of conservation of energy, we know that the kinetic energy of the train is converted to potential energy as it approaches the top of the hill -- causing it to slow down. We also know that the velocity of the train will be lowest when the center of gravity is highest... and the velocity will be highest when the center of gravity is lowest.

Figure 12-D depicts a graph of the velocity of the train vs. the track pathlength, L.

Figure 12-D.   Velocity of a Roller Coaster Train vs. Track Pathlength Over a Hill.

For this discussion, we don't need to know the actual numerical values of the velocity... only the relative magnitudes are needed. The two plots depict the Front-Car ride, and the Back-Car ride.

First, notice that the two cases have the same maximum speed, same minimum speed, and same average speed.

However, for the Front-Car ride, the velocity goes from its highest level at point A, to its lowest level at point B, to an intermediate level at point C. Therefore, it's seen that the riders in the front car experience a net deceleration as they crest the hill.

On the other hand, the Back-Car ride begins at an intermediate speed at position A, drops to its lowest speed at position B, and then reaches its highest speed at position C. Therefore, the riders in the back car experience a net acceleration as they crest the hill.

Thus... there is a real difference between the front and back cars. Even though the minimum and maximum speeds are the same, the riders in the back car experience an acceleration as they go over the top of the hill. This acceleration has a physiological effect on the riders... causing them the feel as if they are being "whipped" over the top.

This "whipping action" is most pronounced on wood roller coasters -- whose cars will tend to lift up from the track since the wheels are not as solidly "locked" to the tracks as they are with steel roller coasters. Hence, many roller coaster enthusiasts -- those who are serious about getting their thrills -- prefer to ride in the back of the train.

Alternatively... there are other reasons that make the ride in the front car highly desirable. In particular, with today's state-of-the-art roller coasters -- with maximum speeds of 60 to 90 mph... or higher -- there are few things that can quite match the wind-blown thrill, and exhilarating view, as you drop down the first hill of a 300 ft tall monster.

So... the bottom line:   What's better?   Front or back???

I say ride the roller coaster at least twice -- once in the front, and once in the back. Then... you decide for yourself.

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