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## Homework Statement

An object with mass m start its journey on x-axis at point (-l,0) with velocity v.

The object feels force which depends on the object's distance to origin.

After T second it reaches the origin. What is T?

## Homework Equations

x(t=0)=-l

v(t=0)=v

f(x)=-kx

x(t=T)=0

T=?

## The Attempt at a Solution

1: dx/dt = v(t)

2: dt = dx/v(t) [from 1]

3: dv/dt = a(t) = (-k/m) * x(t)

4: dt = m * dv/(-k * x(t)) [from 3]

5: v(t) * dv = (-k/m) * x(t) * dx [put 2 to 3]

6: integrating both sides of 5 and putting the values x=0 and the velocity at the origin (which can be calculated by the energy change with integrating f(x) dx from -l to zero) gives the relation between x and v.

7: Putting x in terms of v (found in 6) to 4 and integrating right side from initial velocity to velocity at the origin gives a time (dt at the left side).

But i am not sure about the answer. Any ideas will be great.

Thank you.