The question asks to find the time taken for an object of mass m to fall from a distance from the Earth of R_2 to R_1, where R_1>the radius of the Earth. The mass of the Earth is M and the gravitational constant is G. I got the velocity when the object gets to point R_1, and from there I am stuck.

Original post by blaztroid

How did you get the speed at R_1? Did you use a SUVAT equation? If so, think about which other one might let you solve for time.

I used conservation of energy so I equated the initial potential energy to the final potential plus kinetic

Original post by Javier García

I used conservation of energy so I equated the initial potential energy to the final potential plus kinetic

That's a good idea, and it could help you reach the time by substituting the value for final velocity into another SUVAT equation. But every additional step a chance for a mistake (misreading a value, entering it into your calculator wrong etc.) so often the simplest route is the easiest.

Think about the values you have:

•

s = R_2 - R_1

•

u = 0 (assuming it starts from rest)

•

a = g = 9.81 ms^-2

•

t is the variable to solve for

By equating the potential and kinetic, you obtain another value, v, so you're now free to use any of the equations, but remember it introduces a little room for error.

Hope that helps

Original post by Javier García

The question asks to find the time taken for an object of mass m to fall from a distance from the Earth of R_2 to R_1, where R_1>the radius of the Earth. The mass of the Earth is M and the gravitational constant is G. I got the velocity when the object gets to point R_1, and from there I am stuck.

Original post by mqb2766

Can you assume gravitational acceleration is constant? It sounds like youll have to integrate the usual GM/r^2 appropriately.

Original post by blaztroid

That's a good idea, and it could help you reach the time by substituting the value for final velocity into another SUVAT equation. But every additional step a chance for a mistake (misreading a value, entering it into your calculator wrong etc.) so often the simplest route is the easiest.

Think about the values you have:

In suvat knowing any 3 variables is sufficient to obtain the other two. Here we're dealing with a system in s, u, a, t so choose the one "without v"; s = ut + 0.5at^{2} . And that should be sufficient to solve.

By equating the potential and kinetic, you obtain another value, v, so you're now free to use any of the equations, but remember it introduces a little room for error.

Hope that helps

Think about the values you have:

•

s = R_2 - R_1

•

u = 0 (assuming it starts from rest)

•

a = g = 9.81 ms^-2

•

t is the variable to solve for

By equating the potential and kinetic, you obtain another value, v, so you're now free to use any of the equations, but remember it introduces a little room for error.

Hope that helps

I also thought about that, but then I realised that acceleration is not constant.

Original post by mqb2766

Can you assume gravitational acceleration is constant? It sounds like youll have to integrate the usual GM/r^2 appropriately.

Nope. I got to v_f, and then, I so on a forum (I admit) that you had to do that bit of the integration, however, I don't understand why. I know v is dR/dt as it is the change of R the variable that approaches the Earth over an infinitesimally small amount of time dt, but don't understand why they do that integral, or wether if it is correct. Also, that part of integration of the LHS, I would not know how to integrate it , or why am I integrating from R_2 to R_1 (I assumed it would be from R_2 to R_1 let me know if it is correct please ).

Original post by javier garcía

This is my working.

Looks like about the right thing to do, but some of the limits are a bit strange. Id do the initial integration/conversation of energy from R_2 -> r and from 0 -> v, that way you have variables (not constant R_1) to integrate in the second part where you do seperation of variables. So youre integrating something like

1/sqrt(1/r - 1/R_2))

you seem to fudge this at the end and cant really read your final expression but the usual result is in terms of arctan, so it may be youre ~right but an identity trnasformation away? If its isaac, maybe post the link?

Edit - just seen the posts you posted after the working. Even if its a bit long, maybe write it up a bit clearer and annotate with questions/your understanding and post that. It will make it easier to check.

(edited 6 months ago)

Original post by mqb2766

Looks like about the right thing to do, but some of the limits are a bit strange. Id do the initial integration/conversation of energy from R_2 -> r and from 0 -> v, that way you have variables (not constant R_1) to integrate in the second part where you do seperation of variables. So youre integrating something like

1/sqrt(1/r - 1/R_2))

you seem to fudge this at the end and cant really read your final expression but the usual result is in terms of arctan, so it may be youre ~right but an identity trnasformation away? If its isaac, maybe post the link?

1/sqrt(1/r - 1/R_2))

you seem to fudge this at the end and cant really read your final expression but the usual result is in terms of arctan, so it may be youre ~right but an identity trnasformation away? If its isaac, maybe post the link?

It is a problem I found on Youtube, but that is essentially the heading, to find the time taken to reach R_1 having fallen from R_2. However, what do you mean by the initial integration? Isn't the conservation of energy a way to do it? Or should I integrate there?

Original post by mqb2766

Looks like about the right thing to do, but some of the limits are a bit strange. Id do the initial integration/conversation of energy from R_2 -> r and from 0 -> v, that way you have variables (not constant R_1) to integrate in the second part where you do seperation of variables. So youre integrating something like

1/sqrt(1/r - 1/R_2))

you seem to fudge this at the end and cant really read your final expression but the usual result is in terms of arctan, so it may be youre ~right but an identity trnasformation away? If its isaac, maybe post the link?

Edit - just seen the posts you posted after the working. Even if its a bit long, maybe write it up a bit clearer and annotate with questions/your understanding and post that. It will make it easier to check.

1/sqrt(1/r - 1/R_2))

you seem to fudge this at the end and cant really read your final expression but the usual result is in terms of arctan, so it may be youre ~right but an identity trnasformation away? If its isaac, maybe post the link?

Edit - just seen the posts you posted after the working. Even if its a bit long, maybe write it up a bit clearer and annotate with questions/your understanding and post that. It will make it easier to check.

This is what I did for the first part, however, I saw on a forum that I had to get the initial GM/R^2 as negative? But I think it is good like this because why would it be negative? Regardless any help would come in handy

Original post by Javier García

It is a problem I found on Youtube, but that is essentially the heading, to find the time taken to reach R_1 having fallen from R_2. However, what do you mean by the initial integration? Isn't the conservation of energy a way to do it? Or should I integrate there?

You get convervation of energy by integrating up the basic a = -GM/r^2 so either just assume it or derive it from scratch. If its a youtube problem, care to share the link and what do they consider to be the answer?

Original post by Javier García

This is what I did for the first part, however, I saw on a forum that I had to get the initial GM/R^2 as negative? But I think it is good like this because why would it be negative? Regardless any help would come in handy

Looks about right, though if youre just using r as the height, then it would make sense to use v as the current velocity (rather than v_f). Helps to think of it as a variable, like r.

For the -G ... Im presuming "out" is positive, then "in" (acceleration / force / ... " will be negative so -G... and the velocity dr/dt should be the negative root as it will be heading inwards.

Original post by mqb2766

Looks about right, though if youre just using r as the height, then it would make sense to use v as the current velocity (rather than v_f). Helps to think of it as a variable, like r.

For the -G ... Im presuming "out" is positive, then "in" (acceleration / force / ... " will be negative so -G... and the velocity dr/dt should be the negative root as it will be heading inwards.

For the -G ... Im presuming "out" is positive, then "in" (acceleration / force / ... " will be negative so -G... and the velocity dr/dt should be the negative root as it will be heading inwards.

This is my working. The integral bit I saw on a forum. This is the forum https://www.quora.com/If-an-object-is-dropped-from-a-height-of-1000-km-how-much-time-will-the-object-take-to-reach-the-ground-if-neglecting-air-resistance. The bit in the bottom that is covered is R_2=50*6400000, M=6*10^24

(edited 6 months ago)

Original post by Javier García

This is my working. The integral bit I saw on a forum. This is the forum https://www.quora.com/If-an-object-is-dropped-from-a-height-of-1000-km-how-much-time-will-the-object-take-to-reach-the-ground-if-neglecting-air-resistance. The bit in the bottom that is covered is R_2=50*6400000, M=6*10^24

You missed out the fact that velocity is the negative root, not the positive one which probably cancels the error that you shoud be integrating from R2 to R1. However the indefinite integral is

https://www.wolframalpha.com/input?i=integrate+with+respect+to+r+1%2Fsqrt%281%2Fr+-+1%2FR%29

and Im too tired to try and match with your answer, but maybe thats for you to do, but Id use the wolfram ans and put the numbers from the quora problem in and check they work, then check the identity and how to transform if it is.

Original post by mqb2766

You missed out the fact that velocity is the negative root, not the positive one which probably cancels the error that you shoud be integrating from R2 to R1. However the indefinite integral is

https://www.wolframalpha.com/input?i=integrate+with+respect+to+r+1%2Fsqrt%281%2Fr+-+1%2FR%29

and Im too tired to try and match with your answer, but maybe thats for you to do, but Id use the wolfram ans and put the numbers from the quora problem in and check they work, then check the identity and how to transform if it is.

https://www.wolframalpha.com/input?i=integrate+with+respect+to+r+1%2Fsqrt%281%2Fr+-+1%2FR%29

and Im too tired to try and match with your answer, but maybe thats for you to do, but Id use the wolfram ans and put the numbers from the quora problem in and check they work, then check the identity and how to transform if it is.

Thanks, greatly appreciated.

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