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While in the barrel of a tennis ball machine, the acceleration $a$ (in $\left.\mathrm{ft} / \mathrm{s}^{2}\right)$ of a ball is $a=90 \sqrt{1-4 t},$ where $t$ is the time (in s). If $v=0$ for $t=0,$ find the velocity of the ball as it leaves the barrel at $t=0.25 \mathrm{s}$

$15 \mathrm{ft} / \mathrm{s}$

Calculus 2 / BC

Chapter 26

Applications of Integration

Section 1

Applications of the Indefinite Integral

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

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So our acceleration equation we're told is a is equal to 90 times 1 -40 to the one half power. Or the squared of 1 -40. They're both the same thing. I just like to denote it 1 -14. So the one half power. So we're also told that our initial velocity is equal to zero or our velocity is equal to zero and T. Is equal to zero. And so the way that we can solve this equation is we're first gonna find the integral of our acceleration equation which is going to be Our velocity equation. So we have the integral of 90 times one minus four T. To the one half power DT We can take out the 90. So we can say this is equal to 90 times the integral of 1 -40 to the one half power. And this is D. T. And so now we just need to find the integral of this. So the integral of this is we add one to the exponents. So we have 1 -40 To the 3/2. Power We divide by 3/2 which is the same as multiplying by 2/3. And then we also have to divide by -4. Since if we were to find the derivative of this we would bring the the exponent down, we would remind us one to the exponents and then we were times by negative four since we have this negative 40 in here. So we need to also divide by negative four. So we can multiply this by negative 1 4th and then this is all multiplied by 90. So this would be equal to 90 times well negative 1/4 times two thirds. We can get rid of a two. So it's just negative one half times one third. So we get negative 90 Times 1 -40 To the 3/2 power divided by six And we can um simplify this a little bit further so negative 90 divided by three is negative 30. So that we would have negative 30 divided by two which is just negative 15. So this is equal to negative 15 Times 1 -40 to the three halfs power. And we're also told that when V is equal to zero or when T is equal to zero V is equal to zero. So this is plus C. We need to remember that. So we're gonna figure out what C. Is equal to By the fact that we know that V is equal to zero and T is equal to zero. So you can say zero is equal to negative 15 times 1 0 which is just one To the 3/2 power plus c. So 123 hours power is just one. So we have C. R. is equal to negative 15 Plus C. So then C. would be equal to 15. Okay so now we have our velocity equation V. Of T. It's equal to -15 times 1 -40 To the 3/2 power plus 15. And so now we can just plug in T. is equal 2.25. To figure out our velocity after .25 seconds. So Via .25 is equal to negative 15 times 1 -1 four times to five. So 3/2 power plus 15. So four times .25 is just one sweet. Have negative 15 times 1 -1 23 hours powers to just be equal to negative 15 times zero 2 3 abs power plus 15 zero times anything is going to be zero and 0-3 of powers also zero. So we're just gonna get this is equal to 15 and this is in feet per second. So our velocity after A quarter of a second or .25 seconds is equal to 15 ft/s.

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