Velocity Question?

Question

I am a little confused on how I should start this problem. I assume you use integrals but I don't know how. The problem is,
The velocity of a moving object at time t is given by v(t) = e^t/((e^t) +1) Find the position s(t) at time if s(0) = 0.

s(t) = ?

Answer 1

The formula v(t) = e^t/((e^t) +1) is an equation of motion. It is a differential equation for s.

ds/dt = e^t/((e^t) +1).

This is slightly demanding, but we can separate the variables and you notice that e^t is the derivative of ((e^t) +1).

So, we get s(t) = ln ((e^t) +1) + C

Initial conditions give the value C = -ln 2.

Final result is s(t) = ln [1/2((e^t) +1)]

Answer 2

e^t at e^0 = 1 so

1/(1+1) = 1/2

Answer 3

s = ∫ e^t / (e^t + 1) dt
s ( t ) = log (e^t + 1) + C
0 = log( 1 + 1) + C
C = - log 2

s ( t ) = log (e^t + 1) - log 2
s ( t ) = log ( (6t + 1) / 2 )