Question

Помогите пожплуйсьа сросноооо олемпиадаКулька масою 1 зазнає лобового пружного зіткнення з нерухомою кулькою масою 2 = 1,9 кг, прикріпленою до стінки пружиною жорсткістю = 16 Н⁄м (див. рисунок). Тертя в системі відсутнє; радіуси куль однакові. Відомо, що 1 > 2. А) За якої умови друге зіткнення кулі 1 знов відбудеться з нерухомою кулею 2? Б) За якого значення друге зіткнення куль відбудеться через = 1 с після першого? Знайдіть відносну швидкість куль у момент перед цим зіткненням (виразіть її через 0).​

Answer 1

Ответ:

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Answer 2

Давайте розглянемо обидва пункти вашого запиту:
А) Для того щоб друге зіткнення кулі 1 знову відбулося з нерухомою кулею 2, потрібно, щоб куля 1 відскочила від стінки так, що її швидкість змінилася і стала від'ємною. Це станеться, якщо куля 1 відскочить від стінки з достатньою великою швидкістю і покине пружину до моменту другого зіткнення.
Б) Для того щоб друге зіткнення куль відбулося через 1 с після першого, ми можемо використовувати закон збереження енергії. Енергія пружини перед першим зіткненням повинна бути рівною енергії руху кулі після першого зіткнення.
Енергія пружини:
Eпружини = 0.5 * k * x^2,
де k - жорсткість пружини (16 Н/м), x - стиснення пружини після першого зіткнення.
Енергія кулі 1 після першого зіткнення:
Eкулі1 = 0.5 * m * v^2,
де m - маса кулі 1 (1 кг), v - її швидкість після першого зіткнення.
Вам потрібно призвести ці дві енергії до рівності і розв'язати для x:
0.5 * k * x^2 = 0.5 * m * v^2.
Після знаходження x ви можете використовувати відношення швидкостей для знаходження відносної швидкості куль. Це буде залежати від значень x і початкової швидкості кулі 1.

Answer 3

Problem Analysis

We are given a system consisting of two balls, with ball 1 having a mass of 1 kg and ball 2 having a mass of 1.9 kg. Ball 2 is attached to a wall by a spring with a stiffness of 16 N/m. The system is frictionless, and the radii of the balls are equal. We need to determine the conditions under which ball 1 will collide again with ball 2 after the initial collision, as well as the value of time it takes for the second collision to occur.

Solution

To solve this problem, we need to analyze the motion of the balls before and after the collision. Let's break down the problem into two parts:

Part A: Conditions for the Second Collision

To determine the conditions under which the second collision between ball 1 and ball 2 will occur, we need to consider the conservation of energy in the system. The total mechanical energy of the system is conserved in the absence of external forces.

Before the collision, ball 1 is moving towards ball 2 with an initial velocity, and ball 2 is stationary. The spring is compressed due to the initial displacement of ball 2.

During the collision, ball 1 will transfer some of its kinetic energy to ball 2, causing ball 2 to start moving. The spring will then start to expand, exerting a force on ball 2 in the opposite direction.

After the collision, both balls will be moving away from each other. Ball 1 will continue to move with a reduced velocity, while ball 2 will move in the opposite direction with an increasing velocity due to the force exerted by the spring.

For the second collision to occur, ball 1 must reverse its direction and start moving towards ball 2 again. This means that the kinetic energy of ball 1 after the collision must be sufficient to overcome the potential energy stored in the spring.

Let's denote the initial velocity of ball 1 as v1 and the final velocity of ball 1 after the collision as v1'. The potential energy stored in the spring is given by (1/2)kx^2, where k is the stiffness of the spring and x is the displacement of ball 2 from its equilibrium position.

Using the conservation of energy, we can write the following equation:

(1/2)m1v1^2 = (1/2)m1v1'^2 + (1/2)kx^2

where m1 is the mass of ball 1.

Since we know that m1 > m2, the initial kinetic energy of ball 1 is greater than the final kinetic energy after the collision. Therefore, for the second collision to occur, the potential energy stored in the spring must be less than the initial kinetic energy of ball 1.

Let's denote the maximum compression of the spring as x_max. At this point, the potential energy stored in the spring is maximum. Therefore, the condition for the second collision to occur is:

(1/2)kx_max^2 < (1/2)m1v1^2

Let's calculate the maximum compression of the spring using the given values:

m1 = 1 kg k = 16 N/m

Using the equation for the maximum compression of a spring, x_max = (m2/m1) * (v1^2/k), we can substitute the values:

x_max = (1.9 kg / 1 kg) * (v1^2 / 16 N/m)

Simplifying the equation, we get:

x_max = 1.9v1^2 / 16

Therefore, the condition for the second collision to occur is:

(1/2) * 16 * (1.9v1^2 / 16)^2 < (1/2) * 1 * v1^2

Simplifying the equation, we get:

1.9v1^2 < v1^2

This inequality holds true for all values of v1. Therefore, the second collision between ball 1 and ball 2 will always occur, regardless of the initial velocity of ball 1.

Part B: Time for the Second Collision

To determine the time it takes for the second collision to occur, we need to consider the motion of ball 1 after the initial collision.

After the collision, ball 1 will move away from ball 2 with a reduced velocity. The force exerted by the spring will cause ball 2 to start moving in the opposite direction. The motion of ball 1 can be described by the equation of motion for a mass-spring system:

m1 * a1 = -k * x

where a1 is the acceleration of ball 1 and x is the displacement of ball 2 from its equilibrium position.

Since the system is frictionless, the acceleration of ball 1 is constant. Therefore, we can integrate the equation of motion to find the displacement of ball 1 as a function of time.

Integrating the equation, we get:

x = (m1 / k) * A * sin(ωt + φ)

where A is the amplitude of the motion, ω is the angular frequency, and φ is the phase constant.

The time period of the motion is given by T = 2π/ω. Therefore, the time it takes for the second collision to occur is half the time period of the motion.

Let's calculate the time period of the motion using the given values:

m1 = 1 kg k = 16 N/m

The angular frequency is given by ω = sqrt(k/m1). Substituting the values, we get:

ω = sqrt(16 N/m / 1 kg) = 4 rad/s

Therefore, the time period of the motion is:

T = 2π/ω = 2π/4 = π/2 s

The time it takes for the second collision to occur is half the time period of the motion:

t = T/2 = (π/2)/2 = π/4 s

Therefore, the second collision between ball 1 and ball 2 will occur after a time of π/4 seconds.

Part C: Relative Velocity of the Balls before the Second Collision

To find the relative velocity of the balls before the second collision, we need to consider the motion of ball 1 just before the collision.

Before the collision, ball 1 is moving towards ball 2 with an initial velocity v1. Ball 2 is stationary, so its velocity is zero.

The relative velocity of ball 1 with respect to ball 2 is given by v_rel = v1 - 0 = v1.

Therefore, the relative velocity of the balls just before the second collision is equal to the initial velocity of ball 1, which is v1.

Summary

To summarize: - The second collision between ball 1 and ball 2 will always occur, regardless of the initial velocity of ball 1. - The time it takes for the second collision to occur is π/4 seconds. - The relative velocity of the balls just before the second collision is equal to the initial velocity of ball 1, which is v1.

Please let me know if anything is unclear or if you have any further questions!