Question

A roller coaster is moving at \(2.00 \mathrm{~m} / \mathrm{s}\) at the top of the first hill \((h=40.0 \mathrm{~m})\). Ignoring friction and air resistance, how fast will the roller coaster be moving at the top of a subsequent hill, which is \(15.0 \mathrm{~m}\) high?

Short answer

Answer: The final speed of the roller coaster at the top of the second hill is approximately 22.2 m/s.

Step-by-step solution

Define the variables

The given variables are: Initial speed of roller coaster, \(v_1 = 2.00 ~m/s\) Initial height of roller coaster, \(h_1 = 40.0 ~m\) Final height of roller coaster, \(h_2 = 15.0 ~m\) The variable we need to find is the final speed of roller coaster, \(v_2\)

Write down the potential and kinetic energy formulas

We need the formulas for potential and kinetic energy to solve this problem. They are given by: Potential Energy: \(PE = mgh\) Kinetic Energy: \(KE = \frac{1}{2}mv^2\) Where \(m\) is the mass of the roller coaster (unknown but not necessary for this problem as the mass will cancel out), \(v\) is the speed, \(g\) is acceleration due to gravity (approximately \(9.8\,\text{m/s}^2\), and \(h\) is the height.

Calculate the initial total mechanical energy

The initial total mechanical energy at the top of the first hill is the sum of the initial kinetic energy and initial potential energy. Total Mechanical Energy: \(E_{total1} = KE_{initial} + PE_{initial}\) We can use the formulas in step 2 and given variables to find the initial total mechanical energy. \(E_{total1} = \frac{1}{2} m(v_1)^2 + mgh_1\) Since mass \(m\) is constant, we can write simply as: \(E_{total1} = \frac{1}{2}(2)^2 + 9.8(40)\) \(E_{total1} = 2 + 392\) \(E_{total1} = 394\,\text{J}\) (where the unit J stands for joules)

Find potential energy at the top of the subsequent hill

At the top of the subsequent hill, the roller coaster has a potential energy \(PE_{final}\), which can be found using: \(PE_{final} = mgh_2\) \(PE_{final} = 9.8(15)\) \(PE_{final} = 147\,\text{J}\)

Solve for final speed using conservation of total mechanical energy

Since we are ignoring friction and air resistance, the total mechanical energy is conserved. Thus, \(E_{total1} = E_{total2}\). We can use this principle to solve for the final speed at the top of the subsequent hill by finding \(v_2\). \(E_{total2} = KE_{final} + PE_{final}\) \(E_{total2} = \frac{1}{2}m(v_2)^2 + 147\) Using the conservation of total mechanical energy, \(E_{total1} = E_{total2}\) : 394 = \(\frac{1}{2}m(v_2)^2 + 147\) Now, we can solve for \(v_2\): \((v_2)^2 = \frac{2(394-147)}{m}\) \((v_2)^2 = \frac{494}{m}\) \(v_2 = \sqrt{\frac{494}{m}}\) Since mass \(m\) is constant and does not affect the final result, we can finally write: \(v_2 = \sqrt{494}\) \(v_2 \approx 22.2\,\text{m/s}\) The roller coaster will be moving at approximately \(22.2\,\mathrm{m/s}\) at the top of the subsequent hill which is \(15.0\,\mathrm{m}\) high.

Key concepts

Potential Energy

Potential energy, often symbolized as PE, is the energy stored within an object due to its position relative to some other place, such as the ground or the bottom of a hill. It can be likened to the energy that an archer stores in a drawn bow before releasing an arrow.

In the context of a roller coaster, the potential energy is at its maximum when the coaster reaches the top of a hill. It’s calculated using the formula \(PE = mgh\), where \(m\) is the mass of the object, \(g\) is the acceleration due to gravity (9.8 m/s² on Earth), and \(h\) is the height above the reference point.

Thinking about our roller coaster scenario, when the coaster is at the top of the first 40.0-meter hill, its potential energy is calculated based on the height of that hill. As the coaster descends, this potential energy is converted into kinetic energy.

Kinetic Energy

Kinetic energy (KE) is the energy of motion. Any object that is moving has kinetic energy. The greater the mass of the moving object and the faster it's moving, the more kinetic energy it has. The kinetic energy can be calculated using the equation \(KE = \frac{1}{2}mv^2\), with \(m\) being the mass and \(v\) the velocity of the object.

In our roller coaster model, the kinetic energy is highest when the velocity is greatest. As the coaster comes down a hill, it speeds up, thus increasing its kinetic energy. Conversely, as it climbs up the next hill, it loses speed—and thereby loses kinetic energy. But where does this kinetic energy go? It gets transformed back into potential energy as the coaster gains height. This transformation continues back and forth, exemplifying the principle of conservation of mechanical energy, where energy is neither created nor destroyed, only converted from one form to another.

Roller Coaster Physics

The physics of a roller coaster beautifully illustrate the principles of the conservation of mechanical energy. This principle states that, in an isolated system, the total mechanical energy remains constant as long as all the forces doing work are conservative; friction and air resistance are considered non-conservative forces.

In the absence of non-conservative forces like air resistance and friction, as in the textbook problem we've solved, the mechanical energy of a roller coaster—potential and kinetic energy combined—remains constant throughout its ride. So, when the coaster is at the highest point of a hill, it possesses maximum potential energy and minimal kinetic energy. As it zooms downwards, potential energy decreases while kinetic energy increases, maintaining the constant total mechanical energy.

Therefore, when calculating how fast the coaster should be moving at the top of a subsequent, lower hill, we take into account that the sum of kinetic and potential energies must equal the total mechanical energy at the initial point. This is a spectacular real-world demonstration of the fundamental laws of physics in action.