A 1200-kg frictionless roller coaster starts from rest at a height of 24 m. What is its kinetic energy when it goes over a spot that is 12 m high?
PE=€mgh=1200 * 9.8 * 24=282,240 Joules.
KE = 0.
PE + KE = 282,340 J.
PE = 1200 * 9.8 * 12=141,120 J. @ 12 m.
141,120 + KE = 282,240
KE = 282,240 - 141,120 = 141,120 J.
Why did the roller coaster bring a calculator with it when it went over that spot? Because it wanted to calculate its kinetic energy, of course! Alright, here goes the math. We know the height difference between the two spots is 24 m - 12 m = 12 m. Now, let's use a little physics magic to find the kinetic energy. The potential energy of the roller coaster at the first spot is given by mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height. So, at the first spot, the potential energy is 1200 kg * 9.8 m/s^2 * 24 m = 282,240 Joules. Since energy is conserved, this potential energy will be converted to kinetic energy at the second spot. So, the kinetic energy at the second spot is also 282,240 Joules. Voila!
To find the kinetic energy of the roller coaster when it goes over a spot that is 12 m high, we first need to calculate its potential energy at that spot.
The potential energy (PE) of an object at a certain height can be calculated using the formula:
PE = m * g * h
Where:
m = mass of the object
g = acceleration due to gravity (approximated as 9.8 m/s^2)
h = height above the reference point
Given:
m = 1200 kg
h = 12 m
Using the formula, we can calculate the potential energy at the spot:
PE = 1200 kg * 9.8 m/s^2 * 12 m
PE = 141,120 J
Now, to find the kinetic energy (KE) of the roller coaster, we will use the conservation of mechanical energy. Assuming there is no loss of energy due to friction or other factors, the total mechanical energy of the system remains constant.
The mechanical energy (ME) is the sum of the potential energy and the kinetic energy:
ME = PE + KE
Since the roller coaster started from rest at a height of 24 m, its initial mechanical energy (ME_initial) will only consist of potential energy:
ME_initial = PE (at initial height)
ME_initial = 1200 kg * 9.8 m/s^2 * 24 m
ME_initial = 282,240 J
At the spot that is 12 m high, the roller coaster will have transferred some of its potential energy into kinetic energy. Therefore, we can subtract the initial mechanical energy from the potential energy at the spot to find the kinetic energy at that point:
KE = ME_initial - PE_at_spot
KE = 282,240 J - 141,120 J
KE = 141,120 J
Therefore, the kinetic energy of the roller coaster when it goes over a spot that is 12 m high is 141,120 J.
To find the kinetic energy of the roller coaster when it goes over a spot that is 12 m high, we first need to calculate its potential energy at that height and then convert it to kinetic energy.
The potential energy of an object at a certain height is given by the formula:
Potential Energy = Mass * Gravitational Acceleration * Height
In this case, the mass of the roller coaster is 1200 kg, the gravitational acceleration is 9.8 m/s², and the height is 12 m.
Potential Energy = 1200 kg * 9.8 m/s² * 12 m
Next, we need to convert this potential energy into kinetic energy using the principle of conservation of energy. According to this principle, the total mechanical energy of the roller coaster remains constant throughout its motion, assuming no external forces are acting on it (such as friction).
The total mechanical energy is the sum of the potential energy and kinetic energy. Therefore:
Total Mechanical Energy = Potential Energy + Kinetic Energy
Since the roller coaster starts from rest, its initial kinetic energy is zero.
Total Mechanical Energy = Potential Energy + 0
Therefore:
Total Mechanical Energy = Potential Energy
Now, we can calculate the potential energy and find the total mechanical energy:
Potential Energy = 1200 kg * 9.8 m/s² * 12 m
= 141,120 J
So, the total mechanical energy of the roller coaster when it goes over a spot that is 12 m high is 141,120 J.