# Particle Motion. Example Problem 2

Topics: Calculus

## Particle Motion. Example Problem 2

An object moves along the y-axis such that its position is given by y(†) = t^3 – 12t^2 + 21t – 3
at what time does the object change direction?
v(t)=0
y'(t)=3t^2-24t+21=0
t^2-8t+7=0
(t-1)(t-7)=0
t=1, 7
Given an object moving on the y-axis, its position is described by this equation. At what time does
the object change direction?
I believe you will find that the concepts of position, velocity, and acceleration-as they relate to
particles–are fundamental to physics.

We have already worked with this. If an object changes
direction, its velocity must change from positive to negative.
to If you change direction from left to right, you experience a velocity change from positive to
negative, or negative to positive. We've discussed this in previous lessons.
To find the extrema, first take the derivative of a function and set it equal to zero. Then look for :
sign change in the derivative.
IfI said at rest," that would be where the velocity equals zero because I'm not moving.

I'm at rest;
I'm not moving. Furthest to the right–that's a maximum. Furthest to the left–that's a minimum.
You are familiar with all of that. The terminology may differ, but the concepts remain the same.
The derivative of this equation is 3t squared minus 24t plus 21. Setting that equal to zero and
factoring out a three, or dividing through by three, we get St plus 7 equals zero. Dividing through by
t yields solutions of 1 and 7.

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Now we can go ahead and do the number line because it says in "change in direction" at the point
where v changes sign, which means we need a sign change. You should look at a number line for
velocity and see if I get sign changes.
But we can get to the point where we say, "Hey, look. Both exponents are odd for each factor, so I'l1
get sign changes." There are no even exponents, so there will be no bounces at either of those. If
it you're interested in sign charts, it goes positive, negative, positive.
Thus, our final answer is equals and 7. This is where we get the change in direction. We can
also see that if we had said v of t equals 0, it would have just been this as your answer.
The sign work appeared to be unnecessary, but it turned out to be the same for this video question.
That is not always the case.
In summary, the position of an object is the coordinate describing its distance from a fixed point;
velocity is the rate of change in position over time; and acceleration is the rate of change in velocity