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`
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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
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
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