A roller coaster has two hills and starts from rest atop the first hill , which is the higher of the two hills . The secound hill increases if the height of the initial hill is increased by 20m
Your question makes no sense to me.
The roller coaster can not make it to the top of the second hill if the first hill is not higher.
(1/2) m v^2 + m g h = constant
To determine the height of the second hill in relation to the first hill, we need to understand how roller coasters work. Roller coasters use the principle of conservation of energy, specifically the conservation of mechanical energy.
The mechanical energy of an object can be divided into two forms: potential energy (PE) and kinetic energy (KE). Potential energy is the energy an object possesses due to its position or height, and kinetic energy is the energy it possesses due to its motion. In the case of a roller coaster, as the coaster moves from one point to another, the total mechanical energy (PE + KE) remains constant, neglecting energy loss due to friction and air resistance.
Let's assume the initial height of the first hill is h1. Since the roller coaster starts from rest, it has no initial kinetic energy. Therefore, at the top of the first hill, all the initial energy is in the form of potential energy. The total mechanical energy at the top of the first hill is equal to the potential energy:
PE1 = m * g * h1
Where PE1 is the potential energy at the top of the first hill, m is the mass of the roller coaster, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h1 is the initial height of the first hill.
Now, let's say the height of the first hill is increased by 20 meters to h1 + 20. The total mechanical energy at the top of the first hill is still conserved, so we can write:
PE2 = m * g * (h1 + 20)
Where PE2 is the potential energy at the top of the second hill.
To determine the height of the second hill, we need to find the difference between the potential energies of the first and second hills. This can be calculated as follows:
PE2 - PE1 = m * g * (h1 + 20) - m * g * h1
= m * g * 20
We can cancel out the mass and acceleration due to gravity on both sides of the equation, so the height of the second hill (h2) is simply equal to the increased height of the first hill:
h2 = h1 + 20
Therefore, the height of the second hill is increased by 20 meters compared to the height of the initial hill.