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A pyrotechnic rocket is to be launched vertically upwards from the ground. For optimal viewing, the rocket should reach a maximum height of 90 meters above the ground. Ignore frictional forces.(a) How fast must the rocket be launched in order to achieve optimal viewing?(b) Assuming the rocket is launched with the speed determined in part (a), how long after the rocket is launched will it reach its maximum height?

(a) $y=\frac{g}{2} t^{2}+c t$(b) $42.015 \mathrm{~m} / \mathrm{s}$

Calculus 2 / BC

Chapter 1

First-Order Differential Equations

Section 1

Differential Equations Everywhere

Differential Equations

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in this problem. We're trying to design a rocket that needs to reach a maximum height of 90 meters arkels to determine what the initial velocity must be and the time it'll take until it reaches that height of 90 meters. Let's part start by writing down the differential equation that governs the motion that rocket they'll be second derivative of the position. Function equals G. The acceleration due to gravity if we take the first anti derivative will obtain velocity. De y t t is equal to g Times T plus the constant C If we take a further anti derivative. The equation that governs the motion is now why equals G divide by two times T squared plus C Times t plus another concert of integration deep. We need next to determine what the constants C and D would be equal to, and that means we're going to need some initial conditions in this problem. One thing we can assess is that the rocket is going to be launched at ground level. That means that why it at time zero should be zero meters. That gives us our first initial condition. Our goal is also defined. The initial velocity so right that D Y d t must be equal to some quantity that will determine along the further steps of this process. Let's start next with this initial condition. Y a time zero If we input zero in for time, T will give us G over to time zero squared plus C time. Zero plus de, then we'll have zero is equal to D itself. Now let's make that substitution next. Now that motion equation is why is equal to G divide by two times t squared plus c Times T. Now we're going to be able to work out what the initial velocity might be by first, considering the time at which the Rocket is going to reach its initial height. Will say next that the rocket reaches it's max height when its velocity, which is D Y d t, is equal to zero. But D Y t t equals zero brings us back to this equation Here will have zero is equal to g Times T plus C. If we assault this equation for the time T, then T would be equal to negative. See divide by G. So now we know, at least in terms of G and the quantity see the time that the rocket is going to reach its maximum height. But this next equation here tells us what the actual height will be If we plug in this quantity inside T, there will be able to find an expression that allows us to solve for C, so it's work that out very next. I will write that Why at the time, negative see over G by substitution will be G divide by two times negative, see over G quantity squared plus C times negative, see divide by G. But recall also the negative see divide by G is the time at which the rocket reaches its maximum height, which is going to be designed to be 90 meters. But that means this entire left hand side can be replaced by 90 meters and the very next step, so 90 meters will now be equal to. After expanding and simplifying the right hand side. See divide by two G plus negative C squared divide by G. Next, we can obtain common denominators by multiplying the second fraction by 2/2, and this next gives us the equation that 90 is it now equal to C squared, where there should have been a power to here minus two C squared all divide by two G or, in other words, we multiply pull sites by two g, we now have 180 g is equal to negative C squared or the negative C squared is by combining like terms. Let's next multiply both sides by negative one. So that's negative. 180 g is now equal to see and take the square roots off both sides. Excuse me, C squared. So now see is the square root of negative 180 times G in our calculator. This would be approximately equal to the square root of negative 180 times acceleration due to gravity, which is negative 9.807 And that means that sea is approximately equal to 42.15 No rate. To put this information back into our expression for the motion of the object that's going to be fired up and right that why is equal to G over to Times T squared plus 42.15 times t so recall we have two things that we still need to express. We should say what the initial velocity must be, which is D Y d t plus c. Recall that initial velocity is the 42.15 so we can say they're required. Initial velocity is 42.15 meters per second. Next we want to know how long until it reaches 90 meters. But that's the expression that we found here. So the time to reach the max height is T, which is negative. See so negative 42 0.15 Divide by G. But G is negative 9.807 and our calculator that is approximately 4.28 seconds.

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