This has been an issue for me for some time and I am looking to resolve it now.

Question

This has been an issue for me for some time and I am looking to resolve it now. Many times in problems dealing with Dynamics, acceleration is given in terms of position or velocity instead of time. I know how to derive the subsequent velocity and position functions when acceleration is given in time but not the others. I am looking for the method to convert the acceleration function given in position or velocity. Example: a(v) = -2(v^3) m/s to v(t) = ? given v1 and v2 AND a(s) = 3(s^4) + 4(s^3) m/s^2 to v(t) = ? given s1 and s2 I am not looking for a concrete answer, rather the method and steps for the conversion. Any help would be appreciated!

Explanation / Answer

when acceleration is given in terms of v, say, a = f(v), then,just write a = dv/dt

=> dv/dt = f(v)

=> dv/f(v) = dt

further integrate the above expression to obtain v in terms of 't' using limits of v1 and v2

further with v = g(t), you can write it as: ds/dt = g(t)

=> ds = g(t)dt

and integrate to obtain s as function of time.

also, you can differentiate v = g(t) to obtain 'a' as function of time..

b)

when a is given as function of distance, say, a = f(s),

write a = vdv/ds = f(s)

=> v(dv) = (f(s))(ds)

integrate the above to obtain v as a function of s

so, we get v = g(s)

now write v = ds/dt = g(s)

=> ds/g(s) = dt

integrate the above to obtain s as a function of 't'

=> we get s = h(t)

further differentiate s = h(t) to obtain v as function of 't'

further differentiate v as function of 't' to get a as function of 't'..

hope this is what you are looking for and will be helpful..

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