Applications of The Law of Conservation of Energy
The law of conservation of energy states that the sum of the initial kinetic energy, gravitational potential energy, and elastic potential energy will be equal to the final kinetic energy, gravitational potential energy, and elastic potential energy. The formula to demonstrate this concept is KE + GPE + EPE = KE₂ + GPE₂ + EPE₂. This concept is very applicable in real life, whether it is through understanding sports or understanding how certain objects will move. Here are some applications:
Roller Coasters
Roller coasters are a joy to many and are one of the main attractions at amusement parks. They may look very complicated and one may wonder why certain drops are the way that they are. However, this is due to the law of conservation of energy. Throughout the ride, kinetic energy and gravitational potential energy are being converted to one another. When the coaster is at the top of the highest drop, that is when the gravitational potential energy of it will be at its highest. As it goes down the drop, the potential energy will be converted to kinetic energy and as it goes up and down different ramps, the energy from the coaster will be converted to either potential or kinetic energy. This will depend on how far the coaster is off the ground and whether it is on a decline or incline.
One thing that also must be considered is the friction/coefficient of friction caused by the rubbing of the wheels of the car on the track. The law of conservation of energy states that the total amount of energy at one point will be equal to the total amount of energy at another point. In this case, friction will turn useful energy (gravitational potential energy and kinetic energy) into heat energy. This explains why the first drop is always the highest drop of the coaster. A coaster will only go as high as its maximum height from the first drop. Therefore, if the next hill is higher up, the coaster will not have enough kinetic energy to reach the peak of the second hill. As well, if the next hill is close to the height of the original drop, the coaster will still not have enough energy to reach the top of the hill because too much useful energy will have been lost to friction and heat energy. One of the biggest challenges for roller coaster engineers is minimizing friction so that the ride can go on for as long as possible without running out of kinetic energy.
Applying the Math:
Applying the math to roller coasters sounds complicated, but is easier than it might seem. Here is an example of how we can use math to figure out energy conversions. Looking at Figure 1, we will be asked to determine the speed of the cart, as well as the amount of each energy at each point.
We are able to calculate the amount of energy at each point using the law of conservation of energy. For this problem, we will assume that friction is negligible. In this case, we are focusing on kinetic energy and gravitational potential energy. Therefore, our equation will be KE + GPE = KE₂ + GPE₂ or Et = KE + GPE. The formula to calculate kinetic energy is KE = ½ mv² and the formula to calculate gravitational potential energy is GPE = mgh. Therefore, another equation will be ½ mv ²+mgh = ½ mv₂²+mgh₂. As you might notice, there is one common variable amongst every part of the equation: m, or the mass of the cart. This means that the mass of the cart is not necessary to solve these problems. This makes our final equation ½ v²+gh = ½ v₂ ²+gh₂. Now that we have our formula and our given variables, we can plug the given variables in and solve equations.
Point A:
At point A, the cart is at its highest point and has not begun a decline. In this case, it is where the gravitational potential energy will be at its highest and where the kinetic energy will be 0. We are given the height of the cart at 140 metres and our equation:
Et = ½ v²+gh. Substituting the known values into the equation:
Et = ½ (0)2 + (9.8)(140)
Et = 1372 J
Point B:
In this case, we are given the height at point B (95 metres) as well as the total energy of the system from part A (1372 Joules). Knowing all of this, we can calculate the gravitational potential energy at point B, the kinetic energy at point B and the velocity at point B,
Gravitational Potential Energy at point B:
GPE = gh
GPE = (9.8)(95)
GPE = 931 J
Kinetic Energy at point B:
Et = GPE + KE
Et-GPE = KE
1372–931 = KE
KE = 441 J
Velocity at point B:
Et = ½ v²+gh
1372 = ½ v²+ (9.8)(95)
1372–931 = ½ v²
441 = ½ v²
882 = v²
v = 29.6 m/s
Point C:
In this case, we are given the height at point C (80 metres) as well as the total energy of the system from part A (1372 Joules). Knowing all of this, we can calculate the gravitational potential energy at point C, the kinetic energy at point C and the velocity at point C,
Gravitational Potential Energy at point C:
GPE = gh
GPE = (9.8)(80)
GPE = 784 J
Kinetic Energy at point C:
Et = GPE + KE
Et-GPE = KE
1372–784 = KE
KE = 588 J
Velocity at point C:
Et = ½ v²+gh
1372 = ½ v²+ (9.8)(80)
1372–784 = ½ v²
588 = ½ v²
1176 = v²
v = 34.3 m/s
Point D:
In this case, we are given the height at point D (80 metres) as well as the total energy of the system from part A (1372 Joules). However, you may notice that the height at point D is the same as the height in point C. This means that there will be no change in the gravitational potential and kinetic energy from point C to point D. Therefore:
GPE = 784 J
KE = 588 J
v = 34.3 m/s
Pool
Understanding the law of conservation of energy allows us to understand the movement of pool balls and why they react in certain ways when collided with either the edges or other balls. When playing pool, the cue ball is shot at a stationary ball. When the cue ball is hit by the cue stick, a certain amount of kinetic energy is applied from the cue stick to the ball. This kinetic energy will stay mostly the same, due to pool tables having little friction until the cue ball hits the other ball. Once this happens, the energy transfers from the cue ball to the other ball, which causes the other ball to move. The cue ball then loses kinetic energy because the kinetic energy has been transferred to the other ball.
Energy Sources
Understanding the law of conservation of energy allows us to also understand how certain types of energy sources work. For example, hydropower turbines, or free-flow turbines are able to produce electricity using this concept. When water falls from the sky, potential energy is converted to kinetic energy. This kinetic energy is then used to rotate the turbine of a generator in order to produce electricity, which converts the kinetic energy into electrical energy. This concept makes up the gist of hydroelectric energy.
Hydroelectric energy is very important to understand as it is a renewable energy source and if it is available in vast amounts of quantities, is able to become one of the primary sources of electricity. It is able to convert over 90% of available energy into electricity (by comparison, the best fossil fuel plants operate at approximately 60% efficiency) However, this is easier said than done as while there are many advantages to using hydroelectric energy, there are also many disadvantages. For example, although hydropower plants are perceived to be more environmentally friendly than non-renewable sources such as fossil fuels, building these power plants interrupt the natural flow of a river system. This in turn leads to the disruption of animal migration paths, issues with water quality, and human or wildlife displacement. As well, hydropower is ultimately controlled by weather and precipitation trends. This means that the amount of water available for hydropower systems can vary, thus electricity production will also vary. In comparison, forms of energy such as burning fossil fuels will not vary in production unless the supply of these fossil fuels becomes scarce, which although may happen in the future, are not currently happening.
