# Time of motion, natural logarithm help

1. Nov 24, 2009

### SystemTheory

This equation specifies displacement x in terms of motion time t (starting from rest).

$$x = \tau g\left(t + \tau e^{-t/\tau} -\tau\right)$$

where tau = m/b is the system time constant of a mass m suspended from a mechanical braking device with rectilinear damping constant b and g is standard gravity.

Can anyone help me find t as a function of displacement x? I'm having trouble with the logarithm algebra. I'm new to the forum, so any tips are appreciated.

2. Nov 24, 2009

### arildno

This is a trancendental equation for "t", you won't be able to find an exact solution.

3. Nov 24, 2009

### SystemTheory

Thanks for that help.

I'm trying to predict the time down a ramp of known length xL as shown in the diagram. I'd like a closed form approximation to write in a code statement. This will set the transient analysis time based on the input data inside a computer simulation.

The differential equation solved for acceleration looks like this:

$$\frac{dv}{dt} = \frac{mg sin\theta - bv}{m}$$

Any ideas appreciated.

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Last edited: Nov 24, 2009
4. Dec 2, 2009