a-five-foot-tall-boy-tosses-a-tennis-ball-straight-up-from-the-level-of-the-top-of-his-head-neglecti

Question

A five-foot-tall boy tosses a tennis ball straight up from the level of the top of his head. Neglecting frictional forces, the subsequent motion is governed by the differential equation
$$
\frac{d^{2} y}{d t^{2}}=g
$$
If the object hits the ground 8 seconds after the boy releases it, find
(a) the time when the tennis ball reaches its maximum height.
(b) the maximum height of the tennis ball.

Michael Jacobsen · Accepted Answer

for this situation. We&#x27;re told that a boy that&#x27;s FIFA tall tosses a ball and releases right at the height of five feet, giving us the initial condition that why it zero is five feet. We&#x27;re also told that a seconds later the ball hits the ground, giving us this second initial condition. So first, we&#x27;re going to solve this differential equation. By taking the anti derivative of both sides, we will attain that D Y T t is equal to g Times T plus a constant integration. See, ordinarily, we try to solve for C immediately. But note that both of our initial conditions does not involve a derivative. That means the very next step is to take immediately the next anti derivative, which will be why equals from the anti driven of the left hand side G over to Times T&#x27;s word plus C Times T plus a new constant integration. Let&#x27;s call this one upper case D. Now we&#x27;re ready to go to the initial conditions and Saul for both constants C and D. Starting out with D, we can use the initial condition that Wyatt time zero is five feet. Let&#x27;s right out Y at zero by substitution allows us to write G over to time zero squared plus C time zero plus de Now we substitute in the value for Wyatt zero, which is five feet and obtained that five is equal to de itself. Bringing this back in our solution is now why equals g divide by two times t squared plus c Times T plus five. Next, we repeat the process with the second initial condition and right that why at eight equals by substitution g over to times eight squared plus C times eight plus five. Now we make another substitution, or why eight become zero itself from the stated initial condition. So now we have zero is equal to the eight times eight is 64. Divide by two gives us 32 times acceleration due to gravity G plus eight times see plus five and recall. Our goal is to solve for that constant C So subtract five and 32 g so minus five minus 32 g from both sides of the equation is now equal to eight c and divide by eight. So negative five. My is 32 g divide by eight is equal to see itself. One thing we should be very careful about this problem is the units are given in feet. That means G can approximate as negative 32.17 feet per second squared. If we make that approximation will find through a calculator that C is approximately equal to the quantity 128 0.55 So it&#x27;s expressed that next in our solution, we now have that why is equal to G Divide by two T squared minus 128.55 t plus five where that negative came from here and here, which means it should have gone here a swell. So now we have an expression for the motion of this object. We can now answer the question off. When does the ball reach its maximum height? Well, the maximum height will always occur when the derivative is zero. So let&#x27;s take this equation here and set that equal to zero and solve. So for the next phase we have zero is equal to g Times T plus see itself. But we just found out that C is equal to negative 128.55 so our equation now turns into zero is equal to G Times T minus 128.55 If we saw for time T, we&#x27;ll get 128.55 Divide by acceleration due to gravity. G is our time. T. Once we put this in, the calculator will be ready to some rights. Are results will say next that the max height is reached when T is equal to approximately. After throwing this expression to the calculator about 3.98 seconds, then the problem asks for a second question. We want to know what the actual max height would have been. So we take the quantity 3.98 seconds, which tells us when the ball reaches its max height and go back to the formula we saw for why, which tells us the maximum height at any time. T Let&#x27;s evaluate why, at three points 98 which will give us G over to times 3.98 squared minus 128.55 times 3.98 plus five. Upon putting all that work into the calculator, we have that the height at 3.98 seconds, is going to be approximately equal to 259 point 866 feet and this is the max height that was reached.

Averell Hause · Answer

party the instant that the ball leaves the player the player&#x27;s hand until it is caught again. The ball isn&#x27;t free fall, so the acceleration would simply be equaling negative G or simply equaling two negative 9.80 meters per second squared or weaken, say 9.80 meters per second. Squared downwards. Now for part B at at its maximum height, the ball comes to rest momentarily and then begins to fall back downward. So the velocity at the maximum height is actually gonna be equaling zero meters per second. And for part C, we&#x27;re going to say doubt tow. Why equals the why initial t plus 1/2 Ah a T squared. Now here A&#x27;s equaling negative g and the displacement from the ball. When the balls at the throw his hand, the displacement is equaling zero. And so we find that zero equals Avi. Why initial t minus 1/2 g t squared. And so we find that then we can say that here the equation has two solutions t whole zero. That was That would be, of course, when the ball is thrown and then here t equaling to V. Why initial over and G. That would be when the ball is caught. So if the ball is caught, AT T equals 2.0 seconds. We can then say that the why initial is equaling 9.80 meters per second squared G Times T 2.0 seconds divided by two and so v Y initial this equaling 9.80 meters per second. This would be our final answer for Part C and four part D. Then we have velocity. Final squared equals velocity initial square, plus two times the acceleration times Delta y and so here, with philosophy final being zero at the maximum height, we can say that then delta why Max will be equaling essentially negative velocity initial squared divided by two G. So this is equaling negative 9.80 meters per second. Quantity squared, divided by two times negative 9.80 meters per second squared and this is giving us 4.90 meters as the maximum height. That is the end of the solution. Thank you for watching

Anthony Han · Answer

So here we are given information about a certain function. So we have s of T Is equivalent to -16. T sq -8 T plus 64. So when the ball is essentially at the ground, The position will be equivalent to zero ft. So we have to set our S. A. T function equivalent to zero zero equals negative 16 T squared minus 80 T plus 64. So we can divide this by -16. So this will be t squared plus one half tea minus four equals zero. So now we can complete the square, this would be equal to four plus 1/16. So four is 64 divided by 16. So this will be 65 divided by 16. Then we can take the square root of both sides And we can find that t equals negative 1 4th plus or minus the square root of 65 divided by four. We know that we don&#x27;t have negative time. So we have to take the positive square root only. So this would be a negative one plus radical 65. Provided by four. And we take that the time for this process to occur equals 1.77 seconds. So we&#x27;re interested in the velocity at this certain point of time. So for the velocity is the derivative position function and we can find our velocity function Is equivalent to negative 32 T -8. And we have to find this at this specific point in time. And as a result at this specific point in time we find our velocity is equivalent to negative 64.5 meters per second, and this is our final answer.

Samuel Hannah · Answer

So for this problem, we&#x27;ve been us to describe the motion above for on the ball in the air, using Parametric equations from Equation two on page 829 of the textbook, we have that the Parametric equations and project. Our motion could be found using these equations on screen from a question we confined for specific values that gives those of the equations for our case. So from the question we have, the the North is equal to 50. H is equal to six feet. Er is equal to 90. That&#x27;s 90 degrees because we&#x27;re just throwing vertical in the air. And then G is the constant that considers gravity on the ball, and that will take this to be 32. Now we substitute this in for X. We get the X is equal to zero, and this is because the co sign of 90 degrees is simply zero. So this just service the entire extra 20 And this makes sense because we&#x27;re just throwing a ball VerticalNet vertically in the air. And we&#x27;re assuming that there&#x27;s no wind or something that will knock off course on dso you first, something vertical in the air it&#x27;s going to come down in exactly the same place. It started horizontally, so that makes sense. So why we simply substitute in the values to get that Why is equal to minus 16? T squared plus 50 t about six knees off the Parametric equations? The question. Support a now for part B. We&#x27;ve been us to find how long is the bowl is in the air. Andi. To do this, we need to set the initial time on. We&#x27;ll just say that T must be greater than or equal to zero and just start t from zero, which is standard. We just think of time beginning zero on because we&#x27;ve done this. Be time, which the ball is in the air is the same is asking at what time does the ball hit the ground? And we can formulate this more simply by saying for But what time does the vertical distance Why equal zero? Because at that point the ball will stop moving on. This gives us a quadratic equation sold, namely, the zero is equal to minus 16 t squared plus 50 t plus six. We just very ranges by bringing. Your friend gave it to the left hand side. We can use the quadratic formula to solve this where a is equal to 16 b is equal to minus 50 and C is equal to minus six. So the quadratic formula is that X is equal to minus B, which is 50 plus or minus the square root of B squared, which is 2500 minus four times a time. See my six and just noticed that the mind sign of minus four or minus six will cancel each other out, and then this is all divided by two times A, which is 32 on once. We simplify these equations by like doing the subtraction and taking out common factors we get. The X is equal to 25 divided by 16 plus a minus, the square root of 721 divided by 16. There are no X this tea working with tea, so ask the value for time. Now. Time must always be positive. Onda. The solution that takes the minus sign here gives us a negative solution for time. So will discount that and Save A T is equal to 25/16 plus the square root of 721 over 16 which is equal to free 0.241 seconds to free that small places. And that&#x27;s the solution for Part B. Now for Part C. Were asked to find out what maximum height does the bull reach, and we go back to the equation for Why, why is simply a quadratic equation on? If we think about the quadratic equation it has, it&#x27;s like it&#x27;s a parabola, So the maximum point is the vertex of the equation. So the point that flattens off right at the middle and this Vertex is found when t is equal. Tu minus B over two. Way on. We substitute in our values of being a and this is equal to 50 divided by 32 which is 1.56 to 5 seconds. So this is the time at which, for ball that she used its maximum height Onda. We can now find that maximum height by substituting this value of tea into our equation, for why. So why is equal to minus 16 times 1.56 to 5 squared, plus 50 times 1.5 65 plus six Andi Pernis into a car gator. This is equal to 45 north, 6 to 2, that small places, and this will be in feet. So that&#x27;s the maximum height of all reaches in this projectile motion. Now, the last thing to do is to plot the motion of this ball. And what we&#x27;ve done here is we&#x27;ve just posted the vertical displacement as a function of time because the horizontal displacement doesn&#x27;t change. So as we can see, the bull is thrown up, that teeth will zero and it carries on increasing until it reaches this Vertex here, which is that exactly the value we found a 45 point north six on that. At that point, it then decides it doesn&#x27;t want to be in the air anymore. And it starts falling and falling and it hits the ground. At T equals free 60.0.241 Again, just a sui found

Vishal Parmar · Answer

Christian number 73 The height after disagreement is Eun Bi di function s okay now our part eighties while fast is the more moving two seconds after there after being thrown. So we need to take dead already. So your velocity is equally true. Divided the off as an s. These negatives, Ext. Empty square Blessed 64. So, Daddy, with you off discoveries too deep and daddy ready Off the knees one. So here, aggression off the city started to D 64. Now take the music Will do. I do. But you&#x27;re really easy called negative 32 into 64. So here we easy. Quite well, cereal that these after two seconds. The speed of the bullies Ciro Week for a second. Okay, No, But be that is how long after the body strong does it hurt? It&#x27;s maximum height. So for the maximum hide take real music Will do. Siegel here 32 last 64 is Ecuador cereal 30 de is a full 64. So hear music would be go. So that is after TZ cordel through second Wait three days. ISS maximum height. No find that height. So your acidic a little negative. 16 to square last 64 into two. So gooseberries four and 64 into two is 1 20 Hey, so here s is Acquittal 64. Hey, that is hard. Maximum, Right. That is our maximum height. Dropped the ball At T&#x27;s record. Two seconds. Thank you.

Amrita Bhasin · Answer

this word problem asked us. When will the ball be at the highest point in this path? And then if you double your answer, will you find the amount of time, the balls in the air before it hits the court? So for part A, we know the ball is in the highest point at its Vertex. Therefore, we can use the equation negative, be over two A and plug end. We know that this gives us negative 3.8 over to terms negative 4.9, which gives us 0.4 seconds and then we know for part B the answer is gonna be No. We know the ball was thrown a height of 0.5, and if you&#x27;re multiplying 0.5, then you know that it&#x27;s your 0.5 going downwards because you know it&#x27;s going up at a certain amount and then it&#x27;s going down a certain amount

Problem

A pyrotechnic rocket is to be launched vertically…

Problem 25 Hard Difficulty

Answer

a. $g \cong-3.98 \mathrm{} \mathrm{s}^{}$
b. $259.866 \mathrm{ft}$

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a. $g \cong-3.98 \mathrm{} \mathrm{s}^{}$b. $259.866 \mathrm{ft}$

Differential Equations and Linear Algebra

Catherine R.

Heather Z.

Kayleah T.

Michael J.

Integration Review - Intro

Definite vs Indefinite

Stephen W. Goode, Scott A. AnninDifferential Equations and Linear Algebra

Catherine R.

Heather Z.

Kayleah T.

Michael J.

a. $g \cong-3.98 \mathrm{} \mathrm{s}^{}$
b. $259.866 \mathrm{ft}$

Stephen W. Goode, Scott A. Annin

Differential Equations and Linear Algebra