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?

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

inverse v=i*r

nice nice