Final answer:
The speed of the roller coaster at the bottom of an 80.0 m hill is found using conservation of energy, resulting in a speed of approximately 39.6 meters per second.
Step-by-step explanation:
Finding the Speed of a Roller Coaster at the Bottom of a Hill
The speed of the roller coaster at the bottom of an 80.0 m hill can be found using the principles of conservation of energy. We assume that the roller coaster starts from rest, which means that its initial kinetic energy is zero. With no work done by frictional forces, all of the roller coaster's initial potential energy is converted into kinetic energy at the bottom of the hill.
Using the formula for gravitational potential energy (PE = mgh) and kinetic energy (KE = 1/2mv2), and setting them equal due to conservation of energy (PE = KE), we can solve for the final speed (v) of the roller coaster. Since mass (m) cancels out, we only need the height (h) of the hill and the acceleration due to gravity (g = 9.8 m/s2) to find the final speed.
Final speed v = √(2gh) = √(2 ×9.8 m/s2 ×80.0 m) = √(1568 m2/s2) ≈ 39.6 m/s.
Therefore, the roller coaster's speed at the bottom of the hill is approximately 39.6 meters per second.