When acceleration is given as a function of displacement, and you are given the velocity and displacement at a particular time, how do you find the time taken for the particular particle to move a certain distance

Can you give a specific example?

When tackling these problems, first use

a=vdv/dx

Integrating both sides with respect to x to give

v² =f(x)

Taking square root of both sides give

v=g(x)

(Be careful about the sign - use information in the question to get the correct one). Now, as v=dx/dt, we have

dx/dt=g(x)

Separating variables and integrate, we have

$T=\int [1/g(x)]\mathrm{dx}$
with appropriate limits.

The integral, in general, can only be solved numerically.

Hope this helps.

Kerwin