# If you have force depending on distance, then how to find distance depending on time.

This problem has been bothering me for a while now, hope you can help me.

Let's say that the initial velocity of an object, with mass of m is 0 and the initial position is s0 and the force acting on the object is defined as F(s), how do i find the s(t), where t is time. If it's any help, then the F(s) should be periodic. I can also write the exact problem I'm working on, but a general solution would be nice.

rcgldr
Homework Helper

Generally you'll have to translate force into acceleration, this gives you a(s). You can start with a = dv/dt, multply by ds/ds, => a = (dv/ds) (ds/dt) = v dv/ds. This leads to v dv = a(s) ds, which will be the first integration step. You mentioned F(s) is periodic, so take the simple case a(s) = -s, this results in:
v dv = -s ds
1/2 v2 = - 1/2 s2 + c
v = sqrt(c - s2)
ds/dt = sqrt(c - s2)
ds/sqrt(c - s2) = dt
let c = d^2
ds/sqrt(d2 - s2) = dt
sin-1(s / |d|) = t + e (or - cos-1(s / |d|)= t + e)
s = |d| sin(t + e) (or s = -|d| cos(t + e)

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Thank you for your answer. Correct me of I'm wrong, but I understand that d and e are random constants. Isn't there a way to solve it so there wouldn't be any random constants in the answer?

rcgldr
Homework Helper

Thank you for your answer. Correct me of I'm wrong, but I understand that d and e are random constants.
Yes, they are random constants.
Isn't there a way to solve it so there wouldn't be any random constants in the answer?
You need to supply enough initial conditions to solve for the constants, for example, if s(0) = 0 and v(0) = 1, then s(t) = sin(t).

Say i wanted to take definite integrals from both sides of the equation: a(s) ds = v dv, then what should be the intervals for both sides? Should they be equal, or lets say [s1;0] for the left side and [v(s1);v(0)] for the other side?

rcgldr
Homework Helper

Say i wanted to take definite integrals from both sides of the equation: a(s) ds = v dv, then what should be the intervals for both sides? Should they be equal, or lets say [s1;0] for the left side and [v(s1);v(0)] for the other side?
I'm not sure, since this would restrict the equality to defined intervals which could affect the outcome, and you'd still need limits for the ds/sqrt(...) = dt definite integral.

For my example, knowing s(0) and v(0) was enough to solve the example problem. I'm not sure of the advantage of including a second state for s1 earlier on. How would you choose s1?

I know that v(s1) = 0 and i also know how to calculate v(0). But okay, as your way seems better, could you please tell me how you got to the point where s(t) = sin(t). As I'm quite new to all this i didn't really understand how you got rid of those constants. If you could do it step by step, that'd be wonderful. Again, thanks in advance.

rcgldr
Homework Helper

(how to) get rid of those constants.

I used the derived equation for s(t) and it's derivatives, acceleration wasn't used:

s(t) = |d| sin(t + e)
v(t) = |d| cos(t + e)
a(t) = -|d| sin(t + e)

if s(0) = 0 then e = 0 or e = ± π (assuming |d| ≠ 0)
if v(0) = 1 then |d| = 1 and e = 0

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