Question

Решите пожалуйста велосипедист съезжает без начальной скорости с высоты 30 м и делает мертвую петлю радиусом 5 м.Вычислить скорость в верхней точке мертвой петли

Answer 1

Сначала у велосипедиста была энергия потенциальная на высоте 30 метров, а затем в верхней точке петли на высоте 5*2 = 10 метров + кинетическая. Отсюда

$\displaystyle mgH = \frac{mv^2}{2}+mgh\\\\ v = \sqrt{2g(H-h)} = \sqrt{20*20} = 20\text{ m/s}$

Answer 2

Problem Statement

A cyclist starts descending from a height of 30 meters without any initial velocity and performs a vertical loop with a radius of 5 meters. The task is to calculate the velocity at the topmost point of the loop.

Solution

To solve this problem, we can use the principle of conservation of mechanical energy. At the topmost point of the loop, the cyclist will have maximum potential energy and minimum kinetic energy. Therefore, we can equate the potential energy at the starting point to the sum of potential and kinetic energy at the topmost point.

Let's calculate the velocity at the topmost point using the following steps:

1. Calculate the potential energy at the starting point: - The potential energy is given by the formula: potential energy = mass * gravity * height. - Since the mass of the cyclist is not given, we can assume a value of 1 kg for simplicity. - The acceleration due to gravity is approximately 9.8 m/s^2. - The height is given as 30 meters. - Substituting the values, we get: potential energy = 1 kg * 9.8 m/s^2 * 30 m.

2. Calculate the total mechanical energy at the topmost point: - The total mechanical energy is the sum of potential energy and kinetic energy. - At the topmost point, the potential energy is zero (as it is the minimum point) and the kinetic energy is maximum. - Therefore, the total mechanical energy is equal to the kinetic energy at the topmost point. - The kinetic energy is given by the formula: kinetic energy = 0.5 * mass * velocity^2. - Since the mass is assumed to be 1 kg, we can simplify the formula to: kinetic energy = 0.5 * velocity^2.

3. Equate the potential energy at the starting point to the kinetic energy at the topmost point: - Set the potential energy at the starting point equal to the kinetic energy at the topmost point and solve for velocity. - The equation becomes: potential energy = kinetic energy. - Substituting the values, we get: 1 kg * 9.8 m/s^2 * 30 m = 0.5 * velocity^2.

4. Solve for velocity: - Rearrange the equation to solve for velocity: velocity = sqrt((2 * potential energy) / mass). - Substituting the values, we get: velocity = sqrt((2 * (1 kg * 9.8 m/s^2 * 30 m)) / 1 kg).

Now, let's calculate the velocity at the topmost point:

velocity = sqrt((2 * (1 kg * 9.8 m/s^2 * 30 m)) / 1 kg)

Using a calculator, we can evaluate this expression to find the velocity at the topmost point of the loop.

Please note that the above solution assumes ideal conditions, neglecting factors such as air resistance and friction. Additionally, the assumption of a mass of 1 kg for the cyclist is made for simplicity and ease of calculation.

Let me calculate the velocity for you.