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For the following problems, set up and solve the differential equations.You drop a ball with a mass of 5 kilograms out an airplane window at a height of 5000 $\mathrm{m} .$ How long does it take for the ball to reach the ground?

$t_{0}=31.944$ seconds

Calculus 2 / BC

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

Differential Equations

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it's in the positive direction. Deposited y direction in this case is going to be downwards. And we can say that we have our initial position, our initial Y position equaling zero. And then we have y final equaling the why initial t plus 1/2 GT squared again plus direction is downwards. So we have a positive gravity. And so, with the initial velocity equaling zero so we can eliminate that term, we find that the Time T is equal in the square root of two. Why over g? So for part a, we can say that t someone substituting in why someone would be square root of two multiplied by 50 meters, divided by 9.8 meters per second squared. And this is giving us 3.2 seconds approximately. This would be for this part of the motion for the first part of the motion and then for the next part of the motion part of part B, we can say that the displacement here, Weiss obtuse, equaling 100 meters. So here t's up to is gonna be equaling the square root of two multiplied by 100 meters, divided by 9.8 meters per second squared, and this is giving us 4.5 seconds. So the difference between this and the answer to part A is the time required to fall through that 2nd 50 meter distance. So we can say that Delta T is equaling t sub two minus t someone. This would be 4.5 seconds minus 3.2 seconds, equaling 1.3 seconds to fall through second 50 meter distance. So I'll be your answer for part B. That is the end of the solution. Thank you for watching.

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