while-in-the-barrel-of-a-tennis-ball-machine-the-acceleration-a-in-leftmathrmft-mathrms2right-of-a-3

Name: while in the barrel of a tennis ball machine the acceleration a in leftmathrmft mathrms2right of a 3
Uploaded: 2021-06-08
Duration: 4 min 5 s
Channel: Numerade
Description: 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 $t=0.25 \mathrm{s}$

Question

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 $t=0.25 \mathrm{s}$

Kian Manafi · Accepted Answer

So our acceleration equation we&#x27;re told is a is equal to 90 times 1 -40 to the one half power. Or the squared of 1 -40. They&#x27;re both the same thing. I just like to denote it 1 -14. So the one half power. So we&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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.

Wesley Hines · Answer

so we know that the average velocity for any given time period is going to be equal to Displacement X over the displacement of teeth. So to solve this problem for the four different time periods, we&#x27;ll have to come up with the distances, threat and the different times there. So do this. We&#x27;re just gonna make a little tape, so you will have the displacement of tea, each one we&#x27;ll have what time that gives us looking. Then we will get the value of why our X It&#x27;s gonna be why, in this case, because, um, we&#x27;re working with a ball that is in the air. So displacement of X is going to be represented is why in this problem, and we have the displacement of oy and then finally, we can use all of those to give us the velocity at a given point. So the first item had 0.5 seconds. Displacement of T is going to be 0.5. Sense of the time. A team 0.5 seconds is going to be two points. The next item, at 0.1 seconds will be 0.1. Give us two points. One. The third item is 0.5 seconds and this would give us to buy. And then finally, plane 01 seconds. This will give us 2.1. And now we just plug in these four values of T into why? To get the distance. At that time, This will give us zero 13.44 14.76 and 15.7 six. Not a Kightly. The displacement. Why? We&#x27;re just going Teoh, plug all of these values, Um, we&#x27;re gonna subtract 16 from And the reason we&#x27;re subtracting 16 from them is because from why is at equal to tease you would have to, um why equals 16 cause we&#x27;re trying to find the displacement from AT T equals two vs at these four different time points. So we&#x27;re gonna take the times we just came up with why and subtracting from 16. This would give us negative 16 negative 2.56 and negative 1.24 and the negative 0.24 So now, since we know that the is you go to the displacement of X over the displacement of two, we can just plug in our values for the displacement to you in our values. For the displacement of why to solve for T. And this would give us negative. 32 negative, 25.6, negative 24.8 and negative 24.16 And these represents the average velocities or the ball thrown in the air at 0.5 point 105 and 0.1 seconds after t equals two. Now to find the instantaneous philosophy, Um, we could do this graphically. So we notice here that 2.5, it&#x27;s negative. 32 2.1 negative. 25 16 and so on. And we could knows that the closer and closer week que zero seconds, Uh, because instantaneous velocity will be you can view it as if we were to plug in 2.1 So many. It&#x27;s so close to two that its prey, um school knows that as we get closer and closer to two with our time, the velocity keeps decreasing and decreasing it closer to negative 24. And that is actually enough. Um, enough now is this for us to say that the instantaneous velocity as time, bro, just two seconds is negative 24 in this case, feet per second. So you know that the average velocity after two seconds, given those four time points, is this column here and we know that the instantaneous philosophy his native 24 feet percent

Bobby Barnes · Answer

we&#x27;re told a ball is thrown in the air with velocity afford a few per second, and its height can be given by the equation. Why is even 40 team minus 16 t squared? You want to find a velocity when time is acquitted, too. Now the thing will need to recall from physics is that velocity is equal to the derivative or the rate of change of our displacement. So we normally right, this is like Delta lie or don&#x27;t t. But in this case, what we&#x27;re really finding is the derivative of this function active. So if we were to call this function, f a t were wanting to find what is f prime of two. So what we would do for that is the limit as tr purchased two o f t minus Uh uh too All over. T minus two. Now we need to figure out wood after two is so let&#x27;s see. Effort to is going to be so it B a D minus 16 times two squared, which is or which is going to be 64 and then subtracting those should give us 16. So we&#x27;re gonna plug 16 in perfect too. So we&#x27;re gonna have the limit as t approaches to of so effort ease the puncture were given, and I&#x27;m just going to rewrite it like this. Where e. T squared is first. That may have minus 16 all over tea and minus two. Now, if we were to just plug this in directly, we would get 0/0, and I don&#x27;t know what that means. So what we can try to do is defied these two functions to see if they cancel. And I&#x27;m going to do this with synthetic division off on the side here. So it&#x27;d be too on the outside because that&#x27;s the root of T minus two. Read a negative 16 40 negative 16. So dropping the 16 down to that give us negative 32 which would give eight and then two times eight is 16 0 So that means this would be negative 16 x plus, not expert t t plus eight. Which means we can rewrite this us the limit as t approaches to, uh, negative 16 t plus eight. And now we can go ahead and plug to win directly into this, which would give us so negative 16 times two special. Let me just go ahead and write it out first. That&#x27;s gonna be negative. 32 plus eight or negative. 24. And the units with this would be feet per second. So this would be our velocity after two seconds.

Andrea Lee · Answer

the problem. You see Equation 40 team in the 16 T Square as well as the initial velocity of 40 feet per second. The problem asked to find the velocity AT T was too. Which basically means finding the isn&#x27;t thing is really changed. So we plugged in the equation. Uh, institutions may change for one TZ with two we&#x27;ll get be of two is equal to the limit this t goes to of effort T minus effort to over T minus two. Now you have the final one of two is equal to. So if you&#x27;re putting two into the equation, OK, 40 terms to 16 this equal to a T minus 64 which gives you 16. You plug it back into the equation limit t to you. Plug in the equation. 40 T minus 16 t squared, minus 16. No routine rightist, too. And now you factor out the negative eight to help simplify when there is a TV host to negative p times to t square five t close to oh, teen. Now you factor out to t square minus five t plus two a woman t ghost to he to t minus one SWAT team minus two. All their teams to and I can cancel up t minus two. This gets you a little. It has t goes with to open native eight to t minus once. And now, since there&#x27;s nothing else, you&#x27;ve been simplified. What? You to just plug in the to the equation. So too minus one. Then this gets you negative. Eight times. Four minutes? Well, just three. Get you native. 24. He cursed sick.

Anthony Han · Answer

So here we give an example of a certain function where we have Y equals 40 t minus 16 t squared. And were given information that this is our position function, And the derivative of the position function gets the velocity function, which is 40 -32 t. So this is our velocity function And were asked to evaluate the velocity at time of T equals two seconds. So why prime of two would be equivalent to 40 64? So this would be equivalent to negative 24? And our units is specifically in this case feet per second, and this basically shows that one and the ball is thrown into the air at time T equals two seconds. This is giving the part of basically the falling face of the ball, so it&#x27;s after the ball has already reached its peak, and this is our final answer.

Mattie Love · Answer

Yeah, this problem gives you the position equation of a ball that has dropped with state of breast. Your position equation is here as of T equals 4.90 square for your units, your position, yards or meters, your teeth or time units or seconds should report A. You are asked to find the distance that ball travels off the time interval to 2.5 to judicial. We&#x27;re gonna plug two and 2.5 into our position equation and subtract the position that time to position a time 2.5 chills. We asked 2.5 minus s to. So this is going to look like 4.9 times 2.5 squared minus 4.9 timed to square. Now, this is going to come out to be 11 0.0 fire meet two parts be. You&#x27;re asked to find the average velocity over the same time. Interval 2 to 2 point now. Average velocity is the change in position, divided by the change in time. So the change in position is going to be the distance that ball travels over the interpreter to play five, which we just calculated here in part to the top of your fraction is going to be 11.0 chu five meetings and for your denominator do the change in time, which will be 2.5 seconds minus, she said. On this will be to the point to do meeting side by sheer 0.5 seconds, which comes out to 22 0.5 meters per second. And you might also see this written as meters second negative blonde. But it&#x27;s the same thing. And now for part C, you&#x27;re asked to find the instantaneous velocity. Uh, time to second not to do this. You&#x27;re going to need do velocity or be of tea equation to find your velocity equation. You take a driven over your position question. So the choosing come down. Once a club, I felt quite nine to get 9.8. Chief stays in the new expense on the tea is two months. Wonder once you need that. Now that we have are the loss of the equation, we&#x27;re gonna plug the two into it to find her instantaneous velocity. That time too. This is going to be 9.8 turns which was going to give us 19 0.6 meters per second. What&#x27;s it

Problem

A person skis down a slope with an acceleration (…

Problem 11 Easy Difficulty

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 $t=0.25 \mathrm{s}$

Answer

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

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

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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