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18. The gravitational acceleration at the surface of planet X is 11.6 m/s, and the radius of the planet is 66,500 km. An astronaut is hovering at
height h above the planet's surface, and she notices her weight is the same as it is on Earth. What is h?
7000 km
8200 km
5800 km
9400 km

Respuesta :

MathPhys

Answer:

5800 km

Explanation:

Gravitational acceleration is:

g = GM / r²

where G is gravitational constant,

M is the mass of the planet,

and r is the distance from the center.

G and M are constant, so:

g₁ r₁² = g₂ r₂²

Solving for r₂:

r₂ = r₁ √(g₁ / g₂)

When r = 66,500 km, g = 11.6 m/s².

When r = h + 66,500 km, g = 9.8 m/s².

h + 66,500 km = 66,500 km √(11.6 m/s² / 9.8 m/s²)

h + 66,500 km = 72,300 km

h = 5800 km

Based on the information given, the value of the height will be 5800 km.

The gravitational force will be given as:

g = GM/r²

where,

  • G = gravitational constant,
  • M = mass of the planet,
  • r = distance from the center.

Solving for r₂ will be equal to

r₂ = r₁ √(g₁ / g₂)

It should be noted that when r = 66,500 km, g = 11.6 m/s².

Also, when r = h + 66,500 km, g = 9.8 m/s².

Therefore,

h + 66,500 km = 66,500 km × √(11.6 / 9.8)

h + 66,500 km = 72,300 km

Collect like terms

h = 72300km - 66500km

h = 5800 km

In conclusion, the height is 5800km.

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