Enjoy your roller coaster ride in Episode 15! Roller coasters are excellent examples of the law of conservation of energy in action.
Enjoy your roller coaster ride in Episode 15! Roller coasters are excellent examples of the law of conservation of energy in action. (1:17). Visualize the change in mechanical energy type using a bar graph (2:05). We know that the law of conservation of energy states that unless there was external work done on the earth-coaster system, then the total mechanical energy is constant. (2:54) Finally, calculate power, which is how quickly work can be done or how quickly energy is changed. (7:06)
The Question of the Day asks: (8:26) If a roller coaster is located at a position ⅓ as high as the tallest drop (h), using variables only, what would the speed (v) be?
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Hi and welcome to the APsolute RecAP: Physics 1 Edition. Today’s episode will focus on the law of conservation of energy.
Let’s Zoom out:
Unit 4 – Energy
Topic 4.1-4.3
Big Idea – Force Interactions, Change, and Conservation
You are at the amusement park and a friend dares you to ride the tallest roller coaster at the park. The roller coaster train slowly ascends to the top of the hill, then creeps over the peak. You begin your long scream to the bottom of the hill as the train moves faster and faster. The coaster continues rising and falling over additional hills and at times you feel weightless as you leave your seat momentarily. Eventually the brakes of the train slow you to a stop and you are able to find your stomach and realize that you are safe and sound… hopefully.
Let’s Zoom in:
Roller coasters are excellent examples of the law of conservation of energy in action. On a traditional wooden roller coaster, a motor of a particular power rating is used to drag a roller coaster train to the top of a very high hill where it has a large amount of gravitational potential energy. The train hurtles down the hill allowing its mechanical energy to transfer into kinetic energy by the bottom of the hill. The rest of the ride is some combination of gravitational potential energy and kinetic energy conversions eventually ending with work being done to bring the coaster to rest. For the sake of simplifying the roller coaster example, we will assume that the coaster moves without any friction impeding its motion.
Sometimes it is helpful if we use a bar graph to visualize what is going on in terms of energy. Let’s say that our roller coaster has 1,000,000 J of gravitational potential energy when it is at the top of the tallest hill. An energy bar graph shows the total mechanical energy and each of the 3 types of mechanical energy at the time or times being evaluated. For the top of the tallest hill, we would have a bar representing the 1,000,000 J of gravitational potential energy and another bar representing the total mechanical energy that would also be 1,000,000 J. These bars should of course be the same height. There would also be a space for the kinetic energy and the elastic potential energy bars, but they would both be 0 J at the top of the first hill.
Once the coaster makes it to its lowest point, it will have lost all of the gravitational potential energy that it had at the top. Where did the energy go? We know that the law of conservation of energy states that unless there was external work done on the earth-coaster system, then the total mechanical energy is constant. Since there is no friction and air resistance is assumed negligible then there isn’t any work being done on the coaster. All 1,000,000 J of total mechanical energy is in the form of kinetic energy. Sketch that energy bar graph.
How fast would the coaster be traveling if the peak is 100 m tall? You can actually answer this. The coaster had ONLY gravitational potential energy or what I like to call (GPE) at the top, and ONLY kinetic energy (or KE) at the bottom where we are looking for the speed. So, we can set their respective equations equal to each other at those different locations and solve for “v”. You should find that the coaster is traveling with a speed equal to the square root of the quantity 2*g*h or the square root of 2 times earth’s acceleration due to gravity times the height of the drop. You will arrive at a speed of 44.7 m/s, a very reasonable top speed for a roller coaster.
How heavy is the coaster with its riders? That’s easy enough just looking at the highest point and setting the total mechanical energy equal to the GPE at the top. So, 1,000,000 J = mgΔy, and we find the mass is 1000 kg. Always good to know because it could be helpful later.
A sensor at the top of the second, smaller hill monitors the speed of the coaster to be sure it isn’t traveling too quickly. The second hill is only 75% of the height of the first hill. How fast should we expect the sensor to measure that the coaster is traveling? What would the energy bar graph look like? Well, if there is 75% height then the GPE will make up 75% of the total mechanical energy and the remaining 25% will be KE. Don’t believe me? Try it out. Set total mechanical energy or GPE at the highest point equal to the energy forms that exist at the top of hill #2. Both GPE and KE exist there, so mgΔy(hill 1) = mgΔy(hill 2) + ½ m*v2. Once mass is canceled, and some algebra completed you can arrive at an equation for speed again. It should be noted that this speed will absolutely NOT be 25% of the earlier speed since speed is squared in the kinetic energy equation. Using this speed, the mass of the coaster from earlier (1000kg) and the height of the two locations, and voala! You can see that there is 750,000 J of GPE and 250,000 J of KE to make up the remainder of the total mechanical energy that is still constant.
Finally, in order to be able to handle the largest volume of passengers possible, the amusement park would be concerned with how quickly they can get the coaster up the hill so it can traverse the track and fill up with new passengers again. That’s where power comes in! Power is how quickly work can be done or how quickly energy is changed. For our coaster, we will assume it makes it to the top of hill 1 in 30 seconds from its resting location where the riders board. The power is then 1,000,000 J / 30 seconds or 33,333 Watts. In the event that you are in the United States and you know that 746 Watts = 1 horsepower, then that is 44.7 horsepower. The same as the velocity from earlier?!?!? Is that a coincidence? Actually, yes… yes it is.
To Recap…
Total mechanical energy is a combination of gravitational potential energy, kinetic energy, and elastic potential energy. The total mechanical energy is conserved (or constant) as long as no work is done on the system. Power is the change in a system’s energy over time.
Coming up next on the APsolute RecAP Physics 1 Edition, Momentum, Collisions, and their graphs.
Today’s Question of the Day focuses on the work done by a spring.
Question: If a roller coaster is located at a position ⅓ as high as the tallest drop (h), using variables only, what would the speed (v) be?
a) (4/3)gh
b) 2gh
c) (1/3)gh