Question

Мальчик и девочка равномерно движутся по окружностям разных радиусов: r2 = 1,5r1. Во сколько раз скорость движения мальчика должна быть больше скорости движения девочки, чтобы они все время находились на одном радиусе? Во сколько раз будут отличаться ускорения их движений?

Answer 1

Ответ:

1.5 и 1.5

Объяснение:

1) Vм = 1.5Vд, т.к. ему приходится проходить за одно и то же время в 1.5 раза большее расстояние.

2) A = V^2 / R. Поэтому отношение ускорений даст Aм / Aд =  (Vм^2 * Rд) / (Vд^2 * Rм) = 1.5^2 / 1.5 = 1.5

Answer 2

Problem Analysis

We have a boy and a girl moving on circles with different radii. The radius of the second circle is 1.5 times the radius of the first circle. We need to determine the ratio of the boy's speed to the girl's speed in order for them to always be on the same radius. Additionally, we need to find the ratio of their accelerations.

Solution

Let's denote the radius of the first circle as r1 and the radius of the second circle as r2. We are given that r2 = 1.5r1.

To find the ratio of the boy's speed to the girl's speed, we can use the formula for the circumference of a circle:

C = 2πr

The time it takes to complete one full revolution around the circle is the same for both the boy and the girl. Therefore, the ratio of their speeds will be equal to the ratio of the circumferences of their respective circles.

Let's denote the boy's speed as v1 and the girl's speed as v2. The ratio of their speeds can be calculated as:

v1/v2 = (C1/C2)

Substituting the formula for the circumference, we get:

v1/v2 = (2πr1)/(2πr2) = r1/r2

Since we are given that r2 = 1.5r1, we can substitute this value into the equation:

v1/v2 = r1/(1.5r1) = 1/1.5 = 2/3

Therefore, the boy's speed should be 2/3 times the girl's speed in order for them to always be on the same radius.

To find the ratio of their accelerations, we can use the formula for centripetal acceleration:

a = v^2/r

The ratio of their accelerations can be calculated as:

a1/a2 = (v1^2/r1)/(v2^2/r2) = (v1^2/v2^2) * (r2/r1)

Substituting the values we know, we get:

a1/a2 = (v1^2/v2^2) * (1.5r1/r1) = (v1^2/v2^2) * 1.5

Since we know that v1/v2 = 2/3, we can substitute this value into the equation:

a1/a2 = ((2/3)^2/(1/1.5)) * 1.5 = (4/9) * 1.5 = 2/3

Therefore, the ratio of their accelerations is 2/3.

In summary: - The boy's speed should be 2/3 times the girl's speed in order for them to always be on the same radius. - The ratio of their accelerations is 2/3.

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