# Modeling with Linear Equations (DE)

• austinmw89
In summary, the object drops from a height of 20 above a liquid-filled reservoir of depth 50. If the acceleration of gravity is 9.8 and the resistance-to-motion coefficients of air and liquid are 1 and 4, respectively, the time taken by the object to hit the bottom of the reservoir is 22.7 seconds.
austinmw89
1. An object of unit mass is dropped from a height of 20 above a liquid-filled reservoir of depth 50. If the acceleration of gravity is 9.8 and the resistance-to-motion coefficients of air and liquid are 1 and 4, respectively, compute the time taken by the object to hit the bottom of the reservoir.

2. ma=mg-rv , or v'+(r/m)v=g , also the book gives an answer of t= approx. 22.7, but I can't figure out how to get to this answer. 3. This is a linear equation, so I know I need to use an integrating factor to solve. From an example in the book, after finding the integrating factor they obtain solutions of v(t)=(mg/r)+(v0-(mg/r))e^-(r/m)t, t>0 , and also y(t)=(mg/r)t+(m/r)((mg/r)-v0)((e^-(r/m)t)-1)+y0, t>0. I thought I should split this into two problems first. I tried plugging in the numbers to find the time t when the object reaches the origin (the water) by setting v(t)=0, thinking that I could then plug this time into y(t). The thing is though, I don't think this is correct because the object doesn't really hit zero velocity when it hits the water as it would if it was just hitting the ground. I'm really not sure how to precede from here. Any help or suggestions are greatly appreciated, thanks.

Last edited by a moderator:
It's hard to figure these things out without any units being disclosed.

Nevermind, figured it out! I had to solve y(t) for t then solve v(t) to find the velocity that it hits the liquid at, then solve y(t) with that velocity, then find the time it takes to hit the bottom of the reservoir, then add the two times together.

## What is a linear equation?

A linear equation is an algebraic expression that describes a straight line on a graph. It has the form y = mx + b, where m is the slope of the line and b is the y-intercept.

## What is the purpose of modeling with linear equations?

The purpose of modeling with linear equations is to create a mathematical representation of a real-world situation. This allows us to make predictions and analyze relationships between variables.

## What is the process for creating a linear equation model?

The process for creating a linear equation model involves identifying the variables and their relationships, determining the slope and y-intercept, and writing the equation in the form y = mx + b.

## What are some common applications of linear equation modeling?

Linear equation modeling is commonly used in fields such as economics, physics, and engineering to analyze data and make predictions. Some specific applications include calculating population growth, predicting stock prices, and determining the trajectory of a projectile.

## What are some limitations of linear equation modeling?

Linear equation modeling assumes a linear relationship between variables, which may not always be accurate. It also requires a constant rate of change, which may not hold true in all situations. Additionally, it can only model relationships between two variables at a time.

• Calculus and Beyond Homework Help
Replies
2
Views
488
• Calculus and Beyond Homework Help
Replies
0
Views
573
• Calculus and Beyond Homework Help
Replies
3
Views
784
• Calculus and Beyond Homework Help
Replies
12
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
463
• Calculus and Beyond Homework Help
Replies
1
Views
637
• Calculus and Beyond Homework Help
Replies
8
Views
534
• Calculus and Beyond Homework Help
Replies
23
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
776