Falling object (gravity + kinematics)

[mbrmbrg]

Homework Statement

An object is dropped from an altitude of one Earth radius above Earth's surface. If M is the mass of Earth and R is its radus, find the speed of the object just before it hits Earth.

Homework Equations

$$F_g=ma_g=\frac{GMm}{r^2}$$

$$v^2=v_0^2+2ah$$

The Attempt at a Solution

$$F_g=\frac{GMm}{(2R)^2}=ma_g$$

$$a_g=\frac{GM}{4R^2}$$

Now, plug that into the kintematics equation and get

$$v^2=0+2(\frac{GM}{4R^2})R$$

$$v=\sqrt{\frac{GM}{2R}}$$

But the correct answer is given as $$\sqrt{\frac{GM}{R}}$$, and I can't find my error.

 

[Doc Al]

$$F_g=ma_g=\frac{GMm}{r^2}$$

Note that F_g and a_g are not constant, but are functions of r.

$$v^2=v_0^2+2ah$$

But this kinematic equation assumes constant acceleration.

The Attempt at a Solution

$$F_g=\frac{GMm}{(2R)^2}=ma_g$$

$$a_g=\frac{GM}{4R^2}$$

That's only the acceleration at the point r = 2R; as the object falls, the acceleration increases.

Your error is treating this as a constant acceleration problem. Instead of using kinematics, why not use energy conservation? (What's the general form for gravitational PE? Note that "mgh" is only valid near the Earth's surface--no good here.)

 

[mbrmbrg]

Thanks, got it now!