Question

На аттракционе "Американские горки" есть "мертвая петля" радиусом 10 метров. С какой минимальной высоты над "днем петли" должен начать движение вагончик с людьми, чтобы удержаться на колее в верхней точке петли? Считайте что на самой верхней точке скорость движения равна нулю.

Answer 1

Если высоту мерить от земли, да еще вверхней точке скорость равна нуль, то пропишем закон сохранения энергии:

mgH=mg2R+mv^2/2

где V^2=aR,а-центростремительное. Для минимального случая а=gследовательно H=5mgR/2mg=2.5R

Answer 2

Calculation of the Minimum Height for the Roller Coaster Loop

To calculate the minimum height required for the roller coaster car to stay on the loop, we need to consider the forces acting on the car at the topmost point of the loop. At this point, the car is momentarily at rest, and the only force acting on it is the gravitational force.

The minimum height required can be determined using the following equation:

Centripetal Force = Gravitational Force

The centripetal force is given by the equation:

Centripetal Force = (Mass of the Car) x (Acceleration)

The gravitational force is given by the equation:

Gravitational Force = (Mass of the Car) x (Acceleration due to Gravity)

Setting these two forces equal to each other, we can solve for the minimum height required.

Let's assume the mass of the car is m and the radius of the loop is 10 meters.

The acceleration due to gravity is approximately 9.8 m/s^2.

The acceleration of the car can be calculated using the equation:

Acceleration = (Velocity)^2 / (Radius)

Since the velocity at the topmost point is zero, the acceleration will also be zero.

Substituting the values into the equations, we have:

(m) x (0) = (m) x (9.8)

Simplifying the equation, we find that the mass of the car cancels out, leaving us with:

0 = 9.8

This equation is not possible to solve, as it results in an inconsistency. Therefore, there is no minimum height that will allow the roller coaster car to stay on the loop with a radius of 10 meters and a velocity of zero at the topmost point.

In conclusion, it is not possible for the roller coaster car to stay on the loop with a radius of 10 meters and a velocity of zero at the topmost point, regardless of the initial height from which it starts.