Gravitation Problem, velocity of object on impact

[rzwhipple]

Homework Statement

Object is moving directly toward earth
initial V = 2000 m/s
distance from center of Earth = 8 X 10^{7} m

Determine speed at impact

Homework Equations

g = \frac{GM}{R^{2}}

The Attempt at a Solution

I am guessing I'll need to integrate to get velocity, but am not sure how to proceed. If I integrate with respect to t, what do I do about the R variable which is also changing?

 

[dr_k]

Homework Statement

Object is moving directly toward earth
initial V = 2000 m/s
distance from center of Earth = 8 X 10^{7} m

Determine speed at impact

Homework Equations

g = \frac{GM}{R^{2}}

The Attempt at a Solution

I am guessing I'll need to integrate to get velocity, but am not sure how to proceed. If I integrate with respect to t, what do I do about the R variable which is also changing?

This problem is probably from a Chapter dealing with potential energy functions, where the integration has already been done, to determine the potential energy function. The gravitational force, between these two objects is a conservative force, so all you'll need is conservation of energy, at an initial (i) and final (f) position. Draw a nice picture of this collision, and you won't make any errors for r_initial and r_final.

 

[rzwhipple]

Ahh, thanks this worked nicely, I didn't use the formula for energy since you need both masses, but problem does say to neglect drag so the pot energy is the same initial and final, and mass of the object was in every term so that dropped out. Thanks again!

 

[dr_k]

Ahh, thanks this worked nicely, I didn't use the formula for energy since you need both masses, but problem does say to neglect drag so the pot energy is the same initial and final, and mass of the object was in every term so that dropped out. Thanks again!

This doesn't sound correct. The initial position is, according to the numbers you gave,

\rm r_{initial} = 8 X 10^7 m = \frac{8 X 10^7 m}{6.37 x 10^6 m} R_E = 12.6 R_E , so there is no way that U_initial = U_final.

 

[rzwhipple]

Maybe I was unclear, here is what I did:

\frac{1}{2}mV^{2}_{I} - \frac{GmM}{R_{I}} = \frac{1}{2}mV^{2}_{F} - \frac{GmM}{R_{F}}

V^{2}_{I} - \frac{2GM}{R_{I}} = V^{2}_{F} - \frac{2GM}{R_{F}}

V^{2}_{F} =V^{2}_{I} - \frac{2GM}{R_{I}} + \frac{2GM}{R_{F}}

V_{F} = \sqrt{V^{2}_{I} - \frac{2GM}{R_{I}} + \frac{2GM}{R_{F}}}

plug and chug, voila--you definitely had me on the right track, however, and I think I forgot to mention we were to neglect drag, so therefore energy is conservative--thanks again