Question

После броска вертикально вверх камень возвращается назад через время Т. Найти временной интервал между моментами прохождения камнем середины максимальной высоты при его подъёме и спуске. Сопротивлением воздуха пренебречь

Answer 1

Уравнение движения камня (ось Y вверх)

$y(t) = v_0t-gt^2/2$

Это парабола с ветвями вниз, а значит максимум достигается в вершине:

$t_{\max} = -v_0/(-2g/2) = v_0/g\\ y_{\max} = y(t_{\max}) = v_0^2/2g$

Теперь решим уравнение

$y(t) = y_{\max}/2 = v_0^2/4g\\\\ v_0t-gt^2/2 = v_0^2/4g\\ gt^2-2v_0t+v_0^2/2g = 0\\\\ D/4 = v_0^2 - v_0^2/2 = v_0^2/2\\\\ t_{1,2} = [v_0\pm v_0/\sqrt{2}]/g\\\\ t_2-t_1 = \frac{2v_0}{\sqrt{2}g}$

Само время всего полета находится как больший корень уравнения

$y(T) = 0\\ v_0T-gT^2/2 = 0\\ T = 2v_0/g\\\\ t_2-t_1 =\frac{2v_0}{\sqrt{2}g} = \frac{T\sqrt{2}}{2}$

Answer 2

Finding the Time Interval between the Stone's Ascent and Descent at the Maximum Height

To find the time interval between the stone's ascent and descent at the maximum height, we can use the principles of projectile motion. When a stone is thrown vertically upwards, it follows a parabolic trajectory due to the force of gravity. We can assume that the resistance of the air is negligible in this scenario.

The time it takes for the stone to reach its maximum height is equal to the time it takes for the stone to fall back to its original position. This is because the stone's vertical velocity decreases symmetrically during its ascent and descent.

Let's denote the time it takes for the stone to reach the maximum height as T. The time interval between the moments of passing through the midpoint of the maximum height during the stone's ascent and descent is also T.

Therefore, the time interval between the stone's ascent and descent at the maximum height is T.

Note: Unfortunately, the search results provided by You.com did not contain specific information regarding the time interval between the stone's ascent and descent at the maximum height. However, the principles of projectile motion described above are widely accepted in physics.

If you have any further questions or need clarification, feel free to ask!