instantaneous-speed-of-a-ball-in-physics-it-is-shown-that-the-height-s-of-a-ball-thrown-straight-up-

Question

Instantaneous Speed of a Ball In physics it is shown that the height $s$ of a ball thrown straight up with an initial speed of 96 ft/sec from ground level is
$$s=s(t)=-16 t^{2}+96 t$$
where $t$ is the elapsed time that the ball is in the air.
(a) When does the ball strike the ground? That is, how long is the ball in the air?
(b) What is the average speed of the ball from $t=0$ to $t=2 ?$
(c) What is the instantaneous speed of the ball at time $r ?$
(d) What is the instantaneous speed of the ball at $t=2 ?$
(e) When is the instantaneous speed of the ball equal to zero?
(1) How high is the ball when its instantaneous speed equals zero?
(g) What is the instantaneous speed of the ball when it strikes the ground?

Amy Jiang · Accepted Answer

Okay, so report A We want to know when our ball starts the ground. So that&#x27;s wonderful. And function s entity is equal to go so we can factor out a 16 t from our turn here. Do we get 16 t times a negative T plus 96/16 6 So this is equal to zero. Okay, so we get to solutions when negative t plus six is equal to go and when 16 t is you could video. So this gives us when t is you could to six on when t is equal to zero. Okay. And I&#x27;ll for part B, we went to find or average speed of our ball from TZ federal to Tuz too. So that&#x27;s our change And why over our change. So that&#x27;s gonna be s evaluated. Cuts to minus s evaluated a thorough over a two minus Errol. So this gives us well as evaluated at two. That&#x27;s a negative 16 times to three part two plus 96 times too. So let&#x27;s put that into our calculator, and I get 128 and then we have us evaluated at so So that&#x27;s going to be Joe So we get 128 minus 0/2. So 128 divided by two is going to be equal to 64. And now we want to find our instantaneous rate of change. So that&#x27;s going to be or derivative. So that&#x27;s us prime at sea. That&#x27;s equal to our limits as T approaches some value. See, So we get, um, our function, which is negative 16 t Square plus 96 t And then we&#x27;re going to subject that by, um, our function evaluated at sea. So that&#x27;s going to be minus so plus a 16 c squared minus a 96 c all over t minus sea yet. So for party, we want to find all right derivative 20 is equal to two. So it&#x27;s plug that into our equation. So that&#x27;s going to be equal to the limits as t approaches to of negative 16 teach squared plus 96 t And then we have, um, play in our Steve for you said that 16 times two to the power of to minus 96 times too. So we have mice 128 over t minus to so Let&#x27;s fact there are numerator. Okay, so we can practice into a Let&#x27;s write that down here. The limits as t approaches to of negative 16 times are come T minus four and or a T minus two. So we see that we can cancel out that t minus still and now, using drugs up, we have negative 16 times two minus four. So that&#x27;s going to be an eight of 16 times native to which is equal to 32.

Amy Jiang · Answer

Okay, So for part a, we want to find when are ball strikes The ground said That&#x27;s when our function is equal to go. So we get negative 16 t squared minus 48 t plus 1 60 So let&#x27;s see if we can practice. Okay, so we contracted us into a Seeley. It&#x27;s scared of 16 times a T minus two any t plus five. So that gives us when t plus five is equal to zero or when t minus two is equal to zero. Well, t is it too negative? Five is a negative number. Therefore, we only consider our positive number since we&#x27;re looking at time, which is positive. So we see that t must be quick to. And now, if report me, you want to find our average speed from to use equipped the one or start it&#x27;s rude girl to to you stick with one. So that&#x27;s going to be or change and s over our change in t So we have as evaluated at one minus s evaluated throw over one minutes. So So what&#x27;s us at one. So that&#x27;s going to be negative. 16 minus 48 plus 1 60 I&#x27;ll plug that into my calculator. So I yet 96 minus well s evaluated a thorough. That&#x27;s going to be 1 60 over one. So we got 96/1 60 which is equal to negative. 64. Okay, enough reports. See, we want to find our instantaneous rate of change, but that&#x27;s going to be equal to or derivative. So we have f s prime at T Ranch nudes. But that&#x27;s not not equal to the limits as T approaches t not of our fallen function. Well, that&#x27;s negative. 16 feet squared minus a 40 18 plus 1 60 in minus plus 16 Jeannot squared plus 48 t knots and then minus 1 60 so we can cancel east to turn. And that&#x27;s going to be over. T minus t not so we&#x27;re left with the limits as to you Approaches, do you not? Of negative 16 t squared, minus 40 80 plus 16. Do you not squared and then plus 48 t not over to you minus. Do you not okay? I know for party. We want to find our It&#x27;s instantaneous speed. At T is equal to one. So when? Two years ago to one. What do we have? What is his s prime At one that&#x27;s going to be our limits as T approaches one of our following function and then really replace T not with one. So this is going to re plus 16 plus 48 all over T minus one. So with 16. 48 Well, we at 64 and I Let&#x27;s see if we could talk. There are new morita. Okay, so we see that we can practice into Well, it&#x27;s right. Limit out again. So he impacted us into negative 16 times, T minus one, and then we have t post more. All over. T minus one. We can counsel out or t minus one. And now, using directs up we get that are instantaneous. Rate isn&#x27;t gonna 16 times one plus four. So that&#x27;s going to be negative. 16 times five, which is equal to negative. 80

Joshua Eastwood · Answer

Okay, so we&#x27;re giving massive T and we&#x27;re told that it is minus 16 t squared, plus 90 60 plus four. So we need the velocity because this question&#x27;s gonna ask us about the velocity, and that&#x27;s the derivative of displacement. So we&#x27;re gonna take the route of displacement, and it&#x27;s gonna be minus 32 t plus 96 and that&#x27;s the derivative. And so then the first part of the question says, What is the velocity after two seconds and after four seconds? So we&#x27;re gonna say, What&#x27;s V of two first? So that&#x27;s minus 32 times two plus 96 which is minus 64 plus 96 which gives me 32 feet per second and then we want via four. Okay, which is minus 128 plus 96. And that gives me positive. Sorry, negative. 32 feet per second. Okay, then the last part says that the derivative, they&#x27;re sort of the maximum height happened. So Part B says Max happens when the velocity is zeros. V F T is equal to zero. So I say, When does that happen? So a C minus 32 t plus 96 is equal to zero. That would mean that 96 was equal to 32 tea. And if I divide by 32 that means that I divide each of these by 32. It demeans that t is equal to three. Okay, so I get t to be three seconds. And then the question says, What&#x27;s the actual maximum height? So I don&#x27;t want ass of three. So s of three is minus 16 sometimes three squared, plus 96 times three plus four. Hey, so I&#x27;m gonna go minus 16 times nine, but just minus 144 plus 96 times three just 288 and then possible words. So my final answer, my final displacement. A maximum displacement is 144 plus 448 okay.

Keyan Sheppard · Answer

and this problem, a ball is thrown into the air so it stays in the air for a few seconds and then comes back down. What we want to determine is the velocity and the acceleration at different points in the ball&#x27;s flight. So we&#x27;ll start by determining by evaluating what is the velocity at the very peak. So I&#x27;m gonna call SB is equal to what and how we&#x27;re going to go about figuring this out is we&#x27;re going to actually evaluate the velocity at different points. So how is it moving throughout? So if you imagine each of these balls is just a snapshot, one second apart from each other at the bottom and after one second it&#x27;s gonna travel that much. So it has a positive upward velocity here and it will have an upward velocity there and there and then it goes down. The ball flies operations peak and then fall back down. And what this means is that every time all the time before reaching the peak it&#x27;s going upwards and all the time after reaching the peak, it goes downwards and actually right at the peak, it has a velocity of zero, one moment before or after. It would be moving. But at the very, very peak, it doesn&#x27;t move at all. Just for that one picture, Perfect moment. Now, what we want to determine is the velocity one second before and one second after. And to do this, we&#x27;re gonna have to assess how velocity changes over time and on Earth we have a nice g which is the acceleration due to gravity, gravity always accelerates all objects on the surface of birth At a rate of 9.8 meters per second squared were 9.8 m per second per second. So for every second the velocity of an object will increased towards Earth By 9.8 m/s. So if you&#x27;re at zero and then wait a second, which is this, then we&#x27;ll be at a velocity of nine .8 m/s. So 1 2nd travels 1 2nd later. The velocity changes by 9.8 m/s, One second before the velocity also changes by 9.8 m/s. So We go back, we&#x27;re looking at 9.8 m/s. But here the velocity is upwards. That&#x27;s 9.8 m per second up and this is 9.8 m/s down. And if you want to look at the Intervals, the one second interval every one second interval velocity will change by 9.8 m/s. Well, actually changed downward by 9.8 m/s. So you could say the velocity decreases if downward is your negative direction Now over two seconds. Imagine from here to here, where is the change between used to moments for the ball And it may seem like it doesn&#x27;t change at all because it goes from 9.8 m/s to 9.8 m/s. But that&#x27;s not quite correct as it goes from 9.8 m per second up, 9.8 m/s down and velocity cares about direction. So what actually happens is it goes from 9.8 m per second up. It changes by 9.8 to reach zero and changes by another 9.8 to reach 9.8 m per second down. And that gives you 19.6. So after two seconds, The change in velocity is equal to 9.8 Plus 9.8, Which is equal to 19.6. So after two seconds are largely changes by 19.6 meters per second. The last thing we wanted Remember is that the acceleration of 9.8 m/s. It&#x27;s true for the entire duration of the ball&#x27;s flight. So the acceleration at every point Here isn&#x27;t 9.8 m/s. Right situation here, 9.8 meters per second squared celebration here, 9.8 years per second squared. And that&#x27;s true for every moment of the ball&#x27;s flight, acceleration does not change. It always points downward as well. That&#x27;s very important. It always points towards her. So that&#x27;s all I hope this was helpful. Thank you

Zulfiqar Ali · Answer

but let&#x27;s write down position of bully right? It&#x27;s equal to minus one divided by two times X relation due to gravity times Ta, uh, square Unless we not times D uh, the air by is equal to minus g. Do you hear? Let&#x27;s be not and similarly for will be, Ah, position is given by minus one divided by two times G into de a minus T b old square artless Be not into de a minus TV So re be by then becomes minus energy into t a t minus t b uh, bliss we not? Right. Okay, now let&#x27;s agree, Teoh, these two week regions. So when we create the street regions minus one, divided by two g de A minus TV whole square. Unless we not into the A minus, T B is equal to minus one. Divided by two g into t a square Uh, bliss be not times, uh, t and, uh, minus Well, we have so minus, uh g minus G into T minus d b old square less to be not rt a minus are to be not t b on. And then finally, we have the Marty to be into two g times t a minus G two B minus two Be not is equal to Zito, right? And do e g Teoh is a culture g Teoh be less to be not and d a is equal to do you TB uh, bliss to be not divided by two times acceleration due to gravity Okay, now I&#x27;ll be a by be a Why is equal to minus energy into a to me, not less g tp divided by two g Uh, this we note So be a my is equal to minus one divided by two g Teoh be And maybe by baby by is equal to minus t into Do we not listen? G times t v divided by two g minus t be lis We not so we be lying. Okay, so they come into it here. So we b why is it going to less one divided by two g to be

Ankit Gupta · Answer

in this question, it is given that Ah, ball is thrown vertically upward with an initial velocity off 96 feet per second. That distance, as in feet off the ball from the ground after T seconds is given, is as soft e equals 90. 60 miners, 16 be square. This is a given expression. Now, in the a part off the question, we have to answer that. At what time tea will the ball strike the ground? No, Initially, the distance remains zero. Hence putting 1960 minus 16. T square is equal to zero. If you get 16 Bs Kuhlman and we get six minus t inside the wreck it solving it. We get to values Peak 10 or T equals six. Now take 10 is initial time. That means after six seconds equals six seconds, the ball will strike the ground. Hence, this is the answer for a part in the be part off the question we have hotel. For what time? Tea is the ball more than 1 28 feet above the ground. That means as softy is greater than 128 putting the value off a softie. 1960 minus 16 T Square greater than 128. We&#x27;ve been sold a cynical reality and find out the values off. Be rearranging the terms we get minus 16 80 square plus 90 60 minus 128. Greater than zero novel multiplied with the negative sign on both side of the inequality we get 16 T Square minus 1960 plus 128 less than zero. As the sign off, the inequality will change dividing it with 16 we get T square miners 60 plus eight less than zero now forming the factors we get. Two factors. T minus two and T minus four is less than zero. Now we have to check for what values off t. The product off T minus two anti minus four remains less than zero. We will take different values off tea for both off defectors. Now we will check for different values off T first in stay less than two. When we put any value, which is less than toe in T minus two, you get a negative answer. 40 equals two. We get it as zero for the values between 2 to 4. If you place any value and try checking 40 minus two. You&#x27;re going to get positive 40 minus four. You&#x27;re going to get positive on 40 greater than four. Also, T minus two will be positive. Taking for another one p minus four will be negative. A few place any value less than two B minus four at equal. Toto will also be negative between the values 2 to 4. It will again give you negative results for take ones for it will be zero on 40 greater than four. It will be positive. Now we will check the results multiplying minus and minus will give us. Plus it will be zero here plus minus will give us miners plus multiplied with zero will be zero Answer on positive answer multiplied with positive one will give us positive. Now we have to check for less than zero. That means the negative. So this is the answer that we are looking for. Hence the value off T will be lying between two and four. So the evil lie between 2 to 4. This is the answer

Problem

Instantaneous Speed of a BaII In physics it is sh…

Problem 47 Hard Difficulty

Answer

(a) 6 sec (b) 64 ft/sec (c) (c) $(-32 t+96) \mathrm{ft} / \mathrm{sec}$ (d) $32 \mathrm{ft} / \mathrm{sec}$ (e) $3 \sec$ (f) $144 \mathrm{ft}$ (g) $-96 \mathrm{ft} / \mathrm{sec}$

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(a) 6 sec (b) 64 ft/sec (c) (c) $(-32 t+96) \mathrm{ft} / \mathrm{sec}$ (d) $32 \mathrm{ft} / \mathrm{sec}$ (e) $3 \sec$ (f) $144 \mathrm{ft}$ (g) $-96 \mathrm{ft} / \mathrm{sec}$

Precalculus

Anna Marie V.

Heather Z.

Kristen K.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

Michael SullivanPrecalculus

Anna Marie V.

Heather Z.

Kristen K.

Michael J.

Michael Sullivan

Precalculus