Question

Два шара массами m1= 200 г и m2 =400 г подвешены на нитях длиной l = 67,5 см. Первоначально шары соприкасаются между собой, затем первый шар отклонили от положения равновесия на угол α = 60° и отпустили. Считая удар упругим, определить на какую высоту h поднимется второй шар после удара. Полное решение заранее спасибо)

Answer 1

Первый шар находился на высоте h0 = l cos α, тогда его скорость перед столкновением можно найти через закон сохранения энергии; v^2 = 2gh0.

Пусть первый шар после столкновения полетел в обратную сторону со скоростью u, а второй шар - со скоростью w.
Закон сохранения импульса: m1 v =  m2 w - m1 u
Закон сохранения энергии: m1 v^2 = m2 w^2 + m1 u^2

(m1 u)^2 = (m2 w - m1 v)^2
m1 u^2 = m1 v^2 - 2 m2 wv + m2^2/m1 w^2 - подставляем в ЗСЭ
m1 v^2 = m2 w^2 + m1 v^2 - 2m2 wv + m2^2/m1 w^2
w + m2/m1 w = 2v
w = 2 / (1 + m2/m1) * v

И вновь ЗСЭ, теперь для второго шарика:

h = w^2 / 2g = 2v^2 / g(1 + m2/m1)^2 = 4h0 / (1 + m2/m1)^2

h = 4 * 67.5 * 1/2 / (1 + 2)^2 = 15 см

Answer 2

Problem Analysis

We have two spheres with masses m1 = 200 g and m2 = 400 g, suspended by strings of length l = 67.5 cm. The spheres initially touch each other, and then the first sphere is displaced from its equilibrium position by an angle α = 60° and released. We need to determine the height h to which the second sphere will rise after the collision.

Solution

To solve this problem, we can use the principle of conservation of mechanical energy. The total mechanical energy of the system is conserved before and after the collision.

Before the collision, the total mechanical energy is given by the sum of the potential energy and the kinetic energy:

E1 = PE1 + KE1

After the collision, the total mechanical energy is given by:

E2 = PE2 + KE2

Since the spheres are initially at rest, the initial kinetic energy KE1 is zero. The potential energy PE1 is given by the sum of the potential energies of the two spheres:

PE1 = m1 * g * h1 + m2 * g * h1

where g is the acceleration due to gravity and h1 is the initial height of the spheres.

After the collision, the potential energy PE2 is given by the sum of the potential energies of the two spheres:

PE2 = m1 * g * h2 + m2 * g * h2

where h2 is the final height of the second sphere.

Since the total mechanical energy is conserved, we have:

E1 = E2

Substituting the expressions for PE1 and PE2, we get:

m1 * g * h1 + m2 * g * h1 = m1 * g * h2 + m2 * g * h2

Simplifying the equation, we find:

(m1 + m2) * g * h1 = (m1 + m2) * g * h2

Dividing both sides of the equation by (m1 + m2) * g, we get:

h1 = h2

This means that the final height h2 of the second sphere is equal to the initial height h1 of the spheres.

Therefore, the second sphere will rise to the same height as the initial height of the spheres after the collision.

Answer

The second sphere will rise to the same height as the initial height of the spheres after the collision.