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A ball is rolling such that its velocity $v$ (in $\mathrm{cm} / \mathrm{s}$ ) as a function of time $t$ (in s) is $v=\frac{0.45}{0.25 t^{2}+1} .$ How far does it move in $10.0 \mathrm{s} ?$To find the electric field $E$ from an electric charge distributed uniformly over the entire $x y$ -plane at a distance $d$ from the plane, it is necessary to evaluate the integral $k d \int \frac{d x}{d^{2}+x^{2}} .$ Here, $x$ is the distance from the origin to the element of charge. Perform the indicated integration.

Calculus 1 / AB

Chapter 28

Methods of Integration

Section 6

Inverse Trigonometric Forms

Integrals

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University of Michigan - Ann Arbor

University of Nottingham

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okay in this problem. We've got a table with a ball and the ball rolls off the table and we're told that it's in the air for a time of 0.3 80 seconds. We're told that the velocity at any point the velocity vector is 1.2. I had direction minus 9.8 t in the J hat direction, where I is horizontal and Jay is vertical unnoticed. By the way, this is just an acceleration of 9.8 meters per second. This just means that the starting velocity, if you care, was 1.2 meters per second. And we're told that the displacement vector are is equal to the definite integral from zero to 0.38 seconds of that velocity vector with respect to time. And so, if we solve that, we can say that D R is equal to the integral. From 2.38 of 1.2, I had minus 9.8 t j hat and Weaken treat 1.2 I had as a constant, and we can treat the 9.8 and the J hat as a constant so we can write this as two separate intervals. Delta R is equal to the inter goal of 1.2 I hat from 2.3 eight seconds minus the interval of 9.8 T J hat from zero 2.38 seconds. And so, when we do those intervals, we're gonna go Delta are is equal to this is just a constant and so we're going to multiply it by t should be a hat over that. So it's gonna be 1.2. I had times t evaluated from zero 2.38 and the therapies will just four away minus that t becomes a t squared. So it'll be nine point eight t squared, and it should have put a 1/2 in front. Because as that T square goes, we need to put it. Why half many for the 1/2 and J hat And that also is evaluated from zero 2.38 Now notice that both zeros. When I multiply those three, those will go away. And so my delta are Victor is going to be equal to 1.2 when I plug 2.38 in for tea. That's just going to be 0.38 and that's in the I had direction minus. We've got the 1/2 times 9.8 times 0.38 squared. Let me write those all down 1/2 9.8 0.3 eight squared and that's in the J hat direction. And when we go through and do all the math for those we get, our Delta are is equal to point 456 I had and minus 0.70 eight j hat and that's displacement vector as our object falls off the table, and that's how you saw that from.

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