What is the approximate speed of a roller coaster starting from rest at the top of a 51-meter-high frictionless track at the bottom?

To determine the approximate speed of the roller coaster at the bottom of a frictionless track after starting from rest at a height of 51 meters, we can apply the principle of conservation of energy.

At the top of the track, the roller coaster has potential energy due to its height, which can be calculated using the formula for gravitational potential energy:

Potential Energy (PE) = mgh

where m is the mass of the roller coaster, g is the acceleration due to gravity (approximately 9.81 m/s²), and h is the height (51 meters).

When the roller coaster descends to the bottom of the track, this potential energy is converted into kinetic energy (KE), which is given by the formula:

Kinetic Energy (KE) = 0.5mv²

where v is the speed of the roller coaster at the bottom of the track. Since we are looking for speed, we can set the total energy at the top equal to the total energy at the bottom (assuming no energy is lost):

mgh = 0.5mv².

The mass (m) cancels out from both sides, leading to:

gh = 0.5v².

Now, solving for v gives:

v