Find the object's acceleration each time the velocity is zero (differentiation problem)
At time t, the velocity of an object is given by the function $v = t^2 - 4t +3$
a) Find the object's acceleration each time the velocity is zero.
b) Find the object's velocity each time the acceleration is zero
for part a) should I evaluate $v(0)$ first? then take a derivative of the function?
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$\begingroup$for part a) No, you should be finding $t$ when $v(t)=0$, now take the derivative of your velocity function to yield an acceleration function $a(t)$. Use the $t$ or $t$s from your solution to $v(t)=0$ in your acceleration function to calculate the acceleration at that time. Since $a(t)=2t-4$ and since $v(t)=0$ yields $t=1$, $t=3$, it follows that $a(1)=-2$ and $a(3) = 2$. In total, we have found that the velocity equals zero when t = 1 and t = 3. Using that, and the derivative of the velocity function, we found that the accelerations corresponding to those t values are $-2$ and $2$ respectively.
part b) Find the objects velocity each time the velocity is zero? Are you sure you don't mean: find the velocity each time the acceleration is zero? The first question would be trivial so I will assume you mean the latter.
by part a, $a(t) = 2t-4$. Thus when $t=2$, $a(t) = 0$. Now, $v(2) = -1$. So, when the acceleration is 0 the velocity is -1.
$\endgroup$ 1 $\begingroup$$v(0)$ just gives you the object's initial velocity, which is $3 \space \text{units/s}$. To find when the velocity is zero you want to find all $t$ such that $0 = t^2-4t+3$. There should be two solutions, $t_1,t_2$ as $v(t)$ is a quadratic. However if you find a time that is negative, you must scrap that solution because we do not make sense of negative time in physics. Once you know your time(s) where $v(t_1)=v(t_2)=0,$ then you want to plug those values into the acceleration function of the object. Or, evaluate $v'(t_1)$ and $v'(t_2)$ (again, assuming both $t_1$ and $t_2$ are non-negative).
For $(b)$, are you sure that is stated correctly? It is redundant to say that the object's velocity is zero whenever the object's velocity is zero. I'm guessing you want to find the objects velocity when the acceleration is zero. If that is the case, do the opposite of what you did for part $(a)$.
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