an object travels along a straight line. the function ( v(t) = \frac{t + 1}{t - 1} ) gives the objects velocity, in miles per hour, at time ( t 0 ) hours. write a function that gives the objects acceleration ( a(t) ) in miles per hour per hour.

Question

an object travels along a straight line. the function ( v(t) = \frac{t + 1}{t - 1} ) gives the objects velocity, in miles per hour, at time ( t > 0 ) hours. write a function that gives the objects acceleration ( a(t) ) in miles per hour per hour.

Accepted Answer

# Explanation:
## Step1: Recall acceleration definition
Acceleration $ a(t) $ is the derivative of velocity $ v(t) $, so $ a(t) = v'(t) $.
## Step2: Differentiate $ v(t) = \frac{t + 1}{t - 1} $
Use the quotient rule: if $ v(t) = \frac{u(t)}{w(t)} $, then $ v'(t) = \frac{u'(t)w(t) - u(t)w'(t)}{[w(t)]^2} $. Let $ u(t) = t + 1 $, $ u'(t) = 1 $; $ w(t) = t - 1 $, $ w'(t) = 1 $.
Substitute into quotient rule:
$$
\begin{align*}
v'(t) &= \frac{(1)(t - 1) - (t + 1)(1)}{(t - 1)^2}\
&= \frac{t - 1 - t - 1}{(t - 1)^2}\
&= \frac{-2}{(t - 1)^2}
\end{align*}
$$
So $ a(t) = \frac{-2}{(t - 1)^2} $.

# Answer:
$ \boldsymbol{\frac{-2}{(t - 1)^2}} $

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Question

Explanation:

Step1: Recall acceleration definition

Step2: Differentiate \( v(t) = \frac{t + 1}{t - 1} \)

Answer: