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

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

Step-by-step 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