Find the ratio of maximum height to radius of planet

[Elm956]

Homework Statement

Hint: Disregard any dissipative effects of the
atmosphere of the planet.
A projectile is launched from the surface of a planet (mass M, radius R)
with a launch speed equal to 61 percent of the escape speed for that
planet. The projectile will rise to a maximum height and fall back to the
surface of the planet. What will be the ratio of its maximum height above
the surface to radius of the planet, h/R?

Homework Equations

1/2mv² = -Gmm/R
Vf² = Vi² - 2gh

The Attempt at a Solution

1. 1/2mv² = -Gmm/R
v² = -2Gm/R
v = .61√(2Gm/R)

2. Vf² = Vi² - 2gh
0 = Vi² - 2gh
Vi² = 2gh
Vi = √2gh

3. .61√(2Gm/R) = √(2Gh)
.7442(Gm/R) = 2(Gmh/R²)
.7442Gm = 2(Gmh/R)
.7442 = 2h/R
h/R = .3721

Any thoughts or advice on the incorrect answer? Thanks in advance.

 

[ehild]

You made two serious mistakes:

i. You can not apply the formula Vf² = Vi² - 2gh when the height a projectile reaches is comparable with the radius of the planet.

ii. "g" in the formula above is the free -fall acceleration at the surface of the planet, and entirely different from the universal gravitational constant G.

Use the law of conservation of energy in the gravitational field of the planet.

ehild