Question

Calculate the maximum kinetic energy of a roller coaster with an initial height of 1.3 m, a loop height of 0.8 m, and a hill height of 0.6 m, if the marble has a mass of 0.019 kg?

Answer 1

The maximum kinetic energy of a roller coaster with the given parameters can be calculated by using the conservation of energy principle. The difference in height between the initial position and the lowest point is used, along with the mass of the marble and the acceleration due to gravity, to find the maximum kinetic energy. It comes out to be approximately 0.13037 joules.

Step-by-step explanation:

The question asks us to calculate the maximum kinetic energy of a roller coaster, given its initial and various hill heights, and the mass of a marble representing the coaster. This requires the application of the principle of conservation of energy, particularly the conversion of gravitational potential energy into kinetic energy.

To find the maximum kinetic energy, we first need to determine the point where the roller coaster will have the highest speed. Since the maximum kinetic energy corresponds to maximum speed, this typically occurs at the lowest point of the coaster's path, assuming no energy is lost to friction or air resistance.

The highest potential energy is at the initial height. As the roller coaster descends, the potential energy decreases and is converted into kinetic energy. By calculating the difference in gravitational potential energy between the initial height and the lowest point and equating it to the kinetic energy, we can find the maximum kinetic energy the roller coaster can have.

The formula to calculate the maximum kinetic energy at the lowest point is:

KE_max = m × g × (h_initial - h_lowest),

where m is the mass of the marble (0.019 kg), g is the acceleration due to gravity (9.81 m/s²), h_initial is the initial height (1.3 m), and h_lowest is the height of the lowest point the roller coaster reaches, which in this case would be the hill height of 0.6 m. We subtract the lowest point height from the initial height to find the total height difference.

KE_max = 0.019 kg × 9.81 m/s² × (1.3 m - 0.6 m) = 0.019 kg × 9.81 m/s² × 0.7 m = 0.13037 J.

So, the maximum kinetic energy of the roller coaster is approximately 0.13037 joules.

Answer 2

The maximum kinetic energy of the roller coaster is approximately 0.067 joules.

Step-by-step explanation:

To calculate the maximum kinetic energy of the roller coaster, we need to consider the different heights involved. The maximum height is the initial height of 1.3 m. The formula for calculating kinetic energy is KE = 1/2 * mass * velocity^2. Since the roller coaster starts from rest, all of the initial potential energy is converted to kinetic energy when it reaches the bottom of the first hill, which has a height of 0.6 m. So, we can equate the potential energy at the top to the kinetic energy at the bottom:

PE = m * g * h = KE

m * g * h = 1/2 * m * v^2

Where:

• m is the mass of the marble (0.019 kg)
• g is the acceleration due to gravity (9.8 m/s^2)
• h is the height difference (1.3 m - 0.6 m = 0.7 m)
• v is the velocity at the bottom

Simplifying the equation:

0.7 * 9.8 = 1/2 * v^2

v^2 = (0.7 * 9.8) / 1/2

v^2 = 68.6

v ≈ 8.28 m/s

Finally, we can calculate the maximum kinetic energy using the formula:

KE = 1/2 * m * v^2

= 1/2 * 0.019 kg * (8.28 m/s)^2

≈ 0.067 joules