Question

С какой скоростью будут двигаться шары равной массы после абсолютно неупругого удара, если до удара у них были скорости 3 м/с и 4 м/с, направленные во взаимно перпендикулярных направлениях?

Answer 1

Найдем модуль суммы импульсов двух тел до удара. Т.к. скорости были направлены под уголом 90 градусов, то сумма импульсов будет гипотенуза прямоугольного треугольника с катетами начальных двух импульсов, найдем её по теореме Пифагора: 

P2=9m2+16m2

P2=25m2

P=5m

После удара два шара «склеятся» и будут двигаться как одно тело (т.к. удар не упругий). Значит, масса нового тела 2m. 

Найдем скорость, зная импульс: 

U=P/2m

U=5m/2m

U=2,5 м/c 

Answer 2

Calculation of the Final Velocity after an Inelastic Collision

To calculate the final velocity of two balls with equal mass after an inelastic collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.

Let's denote the mass of each ball as m and their initial velocities as v1 and v2. The balls collide in mutually perpendicular directions.

The total momentum before the collision is given by the sum of the individual momenta:

Total momentum before collision = m * v1 + m * v2.

After the collision, the balls stick together and move with a common final velocity, denoted as V.

The total momentum after the collision is given by the product of the total mass (2m, since the balls stick together) and the final velocity:

Total momentum after collision = 2m * V.

According to the principle of conservation of momentum, these two quantities are equal:

m * v1 + m * v2 = 2m * V.

We can solve this equation to find the final velocity V.

Let's plug in the given values: m = 1 kg, v1 = 3 m/s, and v2 = 4 m/s.

m * v1 + m * v2 = 2m * V

1 * 3 + 1 * 4 = 2 * V

7 = 2V

V = 7/2 = 3.5 m/s.

Therefore, after the collision, the balls with equal mass will move with a final velocity of 3.5 m/s in the direction of the collision.

Please note that the above calculation assumes an idealized scenario of an inelastic collision, where the balls stick together and there is no loss of kinetic energy. In real-world scenarios, some energy may be lost due to factors like friction and deformation.