A particle moves in a straight line so that at any time `t geq0`, its velocity is given by `v=1/(1+t)`. a) show that its distance from fixed point O is given by `x=ln(1+t)` b) show that the particle is moving away from O as `t` increases c) find the acceleration when `t=0`

Expert Answers

An illustration of the letter 'A' in a speech bubbles

You first need to know that velocity is derivation of distance traveled over time, that is



So if we have velocity `v` then distance traveled is integral of velocity

`s(t)=int_0^t v(x)dx`


We have `ln(1+t)` instead of `ln|1+t|` because `t` is always non-negative so `ln|1+t|=ln(1+t)`.


It is easy to see that `s(t)` is monotonically increasing function because `ln(1+t)` is increasing function. In fact all logarithms with base greater than 1 are increasing functions. Formally you can prove it like this:


and since `t>0` we have `s'(t)>0` which means that `s` is monotonically increasing function.


Here you should know that acceleration is derivation of velocity over time (or second derivation of distance traveled over time) that is



So at time `t=0` acceleration is 

`a(0)=-1/(1+0)^2=-1` <--Solution

Approved by eNotes Editorial Team