Question

A projectile has a speed of √ gm/3r directed away from a planet, when it is a distance of 4R from the center of the planet. the planet has mass M and radius R. will this projectile be able to escape from the gravitational attraction of the planet?

Answer 1

Answer:

hat the speed of the rocket is lower than the minimum escape velocity whereby the ship cannot escape the planet

Explanation:

To find out if the projectile can escape the planet, let's find the minimum escape velocity using the concepts of energy.

Starting point. On the planet's surface

            Em∠₀ = k + U = ½ m v² - G mM / R²

End point far away

          Emf = U = - g m M / r²

          Em₀ = Emf

          ½ m v² - G m M / R² = -G m M / r²

          v² = G M (1 / R² -1 / r²)

Let's find the velocity for the height of the rocket r = 4R

          v =√GM (1 / R² - 1/16 R²) = √a GM / R²      0.968

This is the speed to escape planet

 

Let's compare this minimum escape velocity with the given value

             

            v = √GM /R²    1 /√ 3

            v = √GM / R 0.577

We can see that the speed of the rocket is lower than the minimum escape velocity whereby the ship cannot escape the planet