   Chapter 6.4, Problem 31E

Chapter
Section
Textbook Problem

Falling Object In Exercises 31 and 32, consider an object with a mass of 4 kilograms dropped from a of1500 meters, where the air resistance is proportional to the velocity.Write the velocity of the object as a function of time t when the velocity after 3 seconds is approximately 31 meters per second. What is the limiting value of the velocity function?

To determine

To calculate: The velocity function and its limiting value when an object of 4 kilogram mass is dropped from the height of 1500 meters where air resistance and velocity are proportional to each other and the velocity reaches approximately 31 meters per second after 5.

Explanation

Given:

When an object of 4 kilogram mass is dropped from the height of 1500 meters where air resistance and velocity are proportional to each other and the velocity reaches approximately 31 meters per second after 5

Formula Used:

The integration of:

dxx=lnx And xndx=xn+1n+1

Calculation:

Since, air resistance and velocity are proportional to each other so

Consider the equation is;

dvdt+kvm=g

Here v is velocity, g is acceleration due to gravity and k is the proportionality constant.

Substitute km by b:

dvdt+bv=g

Rearrange the equation as follows

dvg+bv=dt

Now, use the separation of variables method,

Integrate both the sides,

dvg+bv=dt1bln|g+bv|=t+cg+bv=Kebt

Where, (K=ebc)

Put c=0 and t=0

g=k

Therefore,

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