Conservation of Energy/Gravity Problem

[tak08810]

Homework Statement

An object is projected straight upward from the surface of Earth with an initial speed of 3.37 km/s. What is the maximum height it reaches?

Homework Equations

G = gravitational constant = 6.673x10^(-11) m^3 / (kg s^2)
M = mass of the Earth = 5.9742x10^24 kg
R = radius of the Earth = 6.378x10^6 m
r = max altitude reached (to be determined)
v = initial velocity of object = 3.37 km/s = 3370 m/s
m = mass of the object (numerical value not needed since it will drop out)

conservation of energy

The Attempt at a Solution

Initial Potential Energy = 0 (height is zero)
Initial Kinetic Energy = 1/2 (3370)^2

Final Potential Energy = r * G (M/(R+r)^2)
Kinetic Energy = 0 (velocity would be zero at maximum altitude)

Solving for this, I get the answer as 56700493 m. However the answer is 6.37×10^5 m, what am I doing wrong?

 

[rl.bhat]

Initial potential energy = 0 (height is zero)

This is not correct. PE of the Earth is measured from the center of the earth.
So the initial potential energy is

$$PE_o = \frac{GMm}{R}$$

 

[tak08810]

Ok that makes sense, thank you!