Final answer:
Using the conservation of energy principle, the potential energy at the top of the hill is converted into kinetic energy at the bottom. The mass cancels out and the final speed of the roller coaster at the bottom is calculated to be approximately 44.7 meters per second.
Step-by-step explanation:
To determine how fast the roller coaster is going when it reaches the bottom, we can make use of the principle of conservation of energy. The potential energy (PE) at the top will convert into kinetic energy (KE) at the bottom, assuming no energy loss due to friction or air resistance.
The potential energy at the top of the hill is given by PE = mass × g × height, where g is the acceleration due to gravity (9.8 m/s2) and height is the elevation of the hill (101 m). Since the roller coaster starts from rest, its kinetic energy at the top is 0.
At the bottom of the hill, all the potential energy will have been converted into kinetic energy, KE = 1/2 × mass × velocity2. By setting the PE equal to the KE, we can solve for the velocity:
mass × g × height = 1/2 × mass × velocity2
Notice that the mass cancels out from both sides, and we are left with:
g × height = 1/2 × velocity2
Rearranging the formula to solve for velocity gives:
velocity = √(2 × g × height)
velocity = √(2 × 9.8 m/s2 × 101 m)
velocity = √(1976 m2/s2 × 101 m)
velocity = √(199576 m2/s2)
velocity ≈ 44.7 m/s
Therefore, the speed of the roller coaster when it reaches the bottom will be approximately 44.7 meters per second.