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While visiting Planet Physics, you toss a rock straight up at 15 m/s and catch it 2.7 s later. While you visit the surface, your cruise ship orbits at an altitude equal to the planet's radius every 250 min .
Part A What is the mass of Planet Physics?
Part B What is the radius of Planet Physics?

Sagot :

Answer:

R = 7.915 10⁶ m,  M = 1.04 10³⁵ kg

Explanation:

Let's start by finding the acceleration of the planet's gravity, let's use the kinematic relations

           v = v₀ - g t

the velocity of the body when it falls is the same for equal height, but it is positive when it rises and negative when it falls

          v = -v₀

         -v₀ = v₀ - g t

          g = 2v₀ / t

          g = 2 15 / 2.7

          g = 11.11 m / s²

I now write the law of universal gravitation and Newton's second law

          F = m a

          G m M / R² = m a

         a = g

          g = G M / R²

           

Now let's work with the cruiser in orbit

         F = ma

acceleration is centripetal

         a = v² / r

         G m M / r² = m v² / r                        (1)

the distance from the center of the planet is

        r = R + h

        r = R + R = 2R

we substitute in 1

        G M / 4R² = v² / 2R

        G M / 2R = v²

The modulus of the velocity in a circular orbit is

         v = d / T

the distance is that of the circle

          d = 2π r

          v = 2π 2R / T

          v = 4π R / T

          G M / 2R = 16pi² R² / T²

          T² = 32 pi² R³ / GM

let's write the equations

             g = G M / R² (2)

             T² = 32 pi² R³ / GM

we have two equations and two unknowns, so it can be solved

let's clear the most on the planet and equalize

             g R² / G = 32 pi² R³ / GT²

              g T² = 32 pi² R

             R = g T² / 32 pi²

let's reduce the period to SI units

           T = 250 min (60 s / 1 min) = 1.5 104 s

let's calculate

             R = 11.11 (1.5 10⁴) ² / 32 π²

           R = 7.915 10⁶ m

from equation 2 we can find the mass of the planet

             M = g R² / G

             M = 11.11 (7.915 10⁶) ² / 6.67 10⁻¹¹

             M = 1.04 10³⁵ kg

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