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For the following problems, set up and solve the differential equations.You throw a ball of mass 2 kilograms into the air with an upward velocity of 8 $\mathrm{m} / \mathrm{s}$ . Find exactly the time the ball will remain in the air, assuming that gravity is given by $g=9.8 \mathrm{m} / \mathrm{s}^{2}$

1.63265306 seconds

Calculus 2 / BC

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

Differential Equations

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University of Nottingham

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All right. So for this question, we have been given the acceleration function. They told us that the acceleration off the ball we're gonna call dysfunction A of T is a constant negative 9.8. So the acceleration doesn't change. That is a constant function. Then we've been asked to find the velocity, functions and the position bump shins. So we know that acceleration is the derivative of velocity. So to find the velocity from the acceleration, we're going to take the integral of the acceleration function. So we're gonna do the integral of our acceleration function. Negative 9.8 tt. And that gives us negatives. 9.8 t plus some constant. We don't know what that constant is, but we can figure it out. We were given some initial conditions for velocity. They said that the initial speed of the goal was 40 meters per second. Sue, when the time was zero. The initial speed Waas 40. So we're gonna use this to help us find our velocity function. We know that the of zero equals 40. So that tells us that negative 9.8 time zero waas. Whatever. See, Iwas is equal to 40 because we know that this is our velocity functions the integral of the, um, acceleration function. So we just have to solve this equation for C so negative 9.8 times zero is just zero. So this just becomes C equals 40. So now we have our velocity function. We'll see. Function is negative. 9.8 t plus 40. Now, we just need to find the position, function or height option. We're gonna call that each of teeth, so we know that the position function is just the integral off the velocity function. Looking at these functions going in this direction, we take derivatives to keep going. But going in this direction we're taking the intervals are doing the opposite of the derivatives. So to find our position function, we're gonna take our velocity function. We know that this position function is equal to the integral of the velocity function, so that is equal to the integral of negative 9.8 t plus 40. So when we do, the integral we get, um, we're gonna use power. So we're gonna take the experiment, we're gonna add one, and then we're gonna divide by that some. So when we do that? We get negative. 4.9 t squared was 40 t and then plus seat. Whatever she is so to find. See, we've been given the another initial condition for our heights. We've been told that when the time is zero, the initial height of the bowl is 1.5 meters. So we're gonna use this to figure out what's he is. So if we know that each of zero equals 1.5 and we know that this is R. H, our height function, we're just gonna plug that in. So we get negative 4.9 times zero swear plus 40 times zero, Let's see, is equal to one boy. So I'm looking at that first term. We're gonna start simplifying things. Zero squared to zero and zero times negative. 00.9 is still zero. So this full term disappears and then 40 times zero is also zero. So this germ disappears and we're just left with C equals 1.5. So then our hft function is just negative. 4.9 tease where US 40 t plus 1.5. So this is how we find our velocity and height functions

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