Non-constant acceleration

[simjor50]

Homework Statement

A object is brought up to a distance of 2*R (R=moon radius) from the moon mass center and dropped. Starting velocity is 0.
Calculate the velocity the object has when hitting the moon surface.
Calculate the time it takes to reach the surface.

Homework Equations

Radius R = 1740000m
Moon mass M = 7.35*10^22kg
gravitational acceleration, g=6.67*10^-11*M/R^2
Conservation of energy in gravitational field.

The Attempt at a Solution

Using the laws of conservation of energy i have managed to calculate the speed when hitting the surface: 1679m/s

The problem now is finding the time it takes. Would have been easy if acceleration was constant, but it isn't!
I tried to calculate it as if acceleration was constant and got 2073seconds. This is probably somwhere near the correct answer, but still it's not 100% correct.
If i have done it right the acceleration varies like this:
http://img31.imageshack.us/img31/3249/grafjd.jpg [Broken]

How can i calculate the time used when the acceleration varies with the distance from the moon?

Thanks for all help :)

 

[RoyalCat]

Start out by writing the equation of motion for the falling mass. Try and find an integral to represent the time of the fall. You'll eventually see that you need to express $$r^2$$ in terms of $$v$$ in order to compute the integral. (Conservation of energy makes this fairly easy!)

The integral is pretty tricky, so if allowed, you might want to use a table in order to do it.

 

[simjor50]

What equations do i have to integrate? I have tried to use all equation of motion and integrate them and i always end up with an extreme high number.

Could you please try to explain this in more detail? I do not understand integrals very well..

 

[lightgrav]

in the Energy Conservation equation, solve for v ...
(like you already did before plugging in values).
Now, since v =dx/dt , we can write a formula for dt.
THAT's what they want you to integrate.

 

[simjor50]

Okay, that helped me a bit further :)
I now have this:
http://img708.imageshack.us/img708/3463/formel.jpg [Broken]

For v i inserted what i used to get the speed when hitting the surface. But what should be inserted for ds here?

I really appreciate you helping me here!

 

[lightgrav]

What's the total Mechanical Energy of this 1kg object?
you need the formula for v at an arbitrary location r
as it falls downward ... sqrt(2/m[E_i - PE(r)])
falling downward, ds is usually called -dr .

 

[RoyalCat]

Don't plug in the numbers so soon! They muck everything up and make it far too messy to solve the integral.

Try and find the time as an integral over some function of the velocity.

Remember that: $$F=ma=m\frac{dv}{dt}$$

You may be able to find the time as a function of displacement, but it is much more simple to find the time as a function of velocity. I suggest that you try the latter.

 

[simjor50]

Now i have v as a function of r:
http://img443.imageshack.us/img443/527/16674494.jpg [Broken]

6.67*10^-11 is the gravitational constant
M is the moon mass
R is the moon radius
v(R) = 1679m/s so this is correct compared to the speed i found earlier.

What to do next? I am completely blank...
Do i need a function for "ds" too? Or can i just insert position "r".

 

[RoyalCat]

Use the definition of acceleration $$a=\frac{dv}{dt}$$ to isolate $$dt$$ and then integrate in order to obtain $$t$$ as a function of velocity (Which in turn gives you the time as a function of displacement just as well.

 

[simjor50]

I think i got it now!

http://img9.imageshack.us/img9/9884/tidd.jpg [Broken]

Does this look correct to you? :)

 

[simjor50]

Anyone that can verify that this is done correctly? :)