What states developed in the Americas, and how did they change over time?

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Over time, when the colonies became a true nation, the name took on the meaning encapsulating everyone in this new nation of America, and slowly other people joined in—from the Spaniards down in Florida and on the West Coast to the French who had come down through the Midwest after establishing fur trading in Canada.

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For time t≥0 , the acceleration of an object moving in a straight line is given by a(t)=sin(t23). what is the net change in velo

Firlakuza [10]

The net change in velocity during the given time interval is equal to -0.665.

<u>Given the following data:</u>

Initial time = 0.75 seconds.

Final time = 2.25 seconds.

Acceleration, $a(t) = sin(\frac{t^2}{3} )$

<h3>How to calculate the net change in velocity.</h3>

Since acceleration is the rate of change of velocity of an object, we would derive an expression for the velocity by differentiating with respect to time:

$V(t) = \frac{d}{dt} a(t)\\\\V(t) = \frac{d}{dt} (sin(\frac{t^2}{3} ))\\\\V(t) = cos( \frac{t^2}{3})(\frac{2t}{3})\\\\V(t) = \frac{2t}{3}cos( \frac{t^2}{3})$

Substituting the time interval, we have:

$V(t) = \frac{2\times 2.25}{3}cos( \frac{2.25^2}{3})- \frac{2\times 0.75}{3}cos( \frac{0.75^2}{3})\\\\V(t) = \frac{4.5}{3}cos( \frac{5.0625}{3})- \frac{1.5}{3}cos( \frac{0.5625}{3})\\\\V(t) =1.5(-0.11643)-0.5(0.98247)$