Question

a certain roller coaster has a section shaped like a circular arc with radius 15 m. if your roller coaster car is on the outside of this loop, what speed must it have for you feel weightless at the top of the arc

Answer 1

For riders to feel weightless at the top of a circular arc with a 15-meter radius, the roller coaster car must travel at approximately 12.12 meters per second.

Step-by-step explanation:

For a roller coaster car to achieve the sensation of weightlessness at the top of a circular arc, the car must be moving at a speed such that the centripetal force provided by the circular path is equal to the gravitational force acting on the car. This occurs when the normal force exerted on the riders becomes zero. The formula to calculate the speed (v) for this condition combines the equations for centripetal force (Fc = m × v² / r) and gravitational force (Fg = m × g), where m is mass, v is velocity, r is the radius of the circular path, and g is the acceleration due to gravity (9.8 m/s²). At the top of the loop, centripetal force is the only force acting in the downward direction, and it must equal the weight of the object (m × g). This gives us m × v² / r = m × g, which simplifies to v² / r = g, since the masses cancel out. Finally, by solving for v, we get v =
`√((rg))`.

For a roller coaster loop with a radius (r) of 15 meters, we substitute r into the equation to find the necessary speed (v) for weightlessness:

v = √(15 m × 9.8 m/s²) = √(147) ≈ 12.12 m/s.

Therefore, the speed required for the roller coaster car to make riders feel weightless at the top of the loop with a 15-meter radius is approximately 12.12 m/s.