Question

1.Вычислите первую космическую скорость для высоты над Землей 500 км2.КосмическийкорабльудаляетсяотЗемли. Какизменитсясилатяготения, действующая со стороны Земли на ракету, при увеличении расстояния до центра Земли в 2 раза?ответ поясните

Answer 1

2 при увеличении расстояния в 2 раза сил тяготения уменьшается в 4 раза Потому что сила тяготения обратно пропорциональна квадрату расстояний
1 Скорость спутника на высоте h  вычисляется по формуле
V= V1* корень квадратный из R/ (R+h)=7,7 км/с
Понятно что V1= 8 км\с - первая космическая скорость у поверхности

Answer 2

Calculation of the First Cosmic Velocity

The first cosmic velocity is the minimum velocity required for an object to achieve a stable orbit around the Earth. To calculate the first cosmic velocity for an object at a height of 500 km above the Earth's surface, we can use the formula:

v = sqrt(G * M / r)

Where: - v is the velocity required for orbit (first cosmic velocity) - G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2) - M is the mass of the Earth (approximately 5.972 × 10^24 kg) - r is the distance between the object and the center of the Earth (radius of the Earth plus the height above the surface)

Let's calculate the first cosmic velocity for an object at a height of 500 km above the Earth's surface.

Step 1: Convert the height above the Earth's surface from km to meters. - 500 km = 500,000 meters

Step 2: Calculate the distance between the object and the center of the Earth. - Radius of the Earth = 6,371 km = 6,371,000 meters - Distance (r) = Radius of the Earth + Height above the surface - Distance (r) = 6,371,000 + 500,000 = 6,871,000 meters

Step 3: Calculate the first cosmic velocity (v). - v = sqrt(G * M / r) - v = sqrt((6.67430 × 10^-11) * (5.972 × 10^24) / 6,871,000)

Let's calculate the value using a calculator or Python:

import math

G = 6.67430e-11
M = 5.972e24
r = 6.871e6

v = math.sqrt(G * M / r)
v

The first cosmic velocity for an object at a height of 500 km above the Earth's surface is approximately 7,660 m/s.

Change in Gravitational Force with Increased Distance

When a spacecraft moves away from the Earth, the gravitational force acting on it decreases as the distance between the spacecraft and the center of the Earth increases. According to Newton's law of universal gravitation, the gravitational force (F) between two objects is inversely proportional to the square of the distance (r) between their centers.

F = G * (m1 * m2) / r^2

Where: - F is the gravitational force - G is the gravitational constant - m1 and m2 are the masses of the two objects - r is the distance between the centers of the two objects

In this case, the mass of the spacecraft is not given, but we can assume it to be much smaller compared to the mass of the Earth. Therefore, we can consider the mass of the spacecraft as negligible (m2 ≈ 0).

Let's consider the gravitational force acting on the spacecraft when it is at a certain distance from the center of the Earth (r1). When the distance is increased to twice the initial distance (2 * r1), the new gravitational force (F2) can be calculated using the formula:

F2 = G * (m1 * 0) / (2 * r1)^2

Simplifying the equation:

F2 = G * m1 / (4 * r1^2)

Comparing the initial gravitational force (F1) and the new gravitational force (F2), we can see that the new force is one-fourth (1/4) of the initial force.

Therefore, when the distance to the center of the Earth is increased to twice the initial distance, the gravitational force acting on the spacecraft will decrease to one-fourth of its initial value.

Please note that the above explanation assumes the mass of the spacecraft is negligible compared to the mass of the Earth. If the mass of the spacecraft is significant, the change in gravitational force would be different and would depend on the masses of both objects.