The acceleration of a particle moving along a straight line is inversely proport

Question

The acceleration of a particle moving along a straight line is inversely proportional to its speed with the constant of proportionality being k. The body starts from the position s=x0i with initial speed v0. <?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /?>

Determine the velocity of the particle as a function of time t, and as a function of position s.

Explanation / Answer

a=k/v ==>dv/dt=k/v ==>v^2/2=kt+C ==>v=sqrt(2kt+C1) ===>ds/dt=(sqrt(2kt+C1) ===>s=(2kt+C1)^3/2 *1(1/3k)+C2 at t=0 s=xoi and v=vo ===> C1=vo^2 C2=x0-vo^3/3k v=sqrt(2kt+vo^2) s=(v^3-vo^3)/3k+x0 v=[3k*(s-xo)+vo^3]^1/3

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