Use calculus to solve Eq. (1.9) for the case where the initial velocity, v ( 0 ) is nonzero.

Question

Want to see more full solutions like this?

Answer 1

Textbook Question

Chapter 1, Problem 1P

Use calculus to solve Eq. (1.9) for the case where the initial velocity, v ( 0 ) is nonzero.

Expert Solution & Answer

To determine

To calculate: The solution of the equation dvdt=g−cmv by the use of calculus if the initial velocity v(0) is non-zero.

Answer to Problem 1P

Solution:

The solution of the equation dvdt=g−cmv is v=v(0)e−(cm)t+mgc(1−e−(cm)t).

Explanation of Solution

Given Information:

The equation, dvdt=g−cmv.

The initial velocity v(0) is non-zero.

Formula used:

Integration property of polynomial,

∫1a+bxdx=1bln(a+bx)+C

Logarithmic property,

lna−lnb=ln(ab)

Calculation:

Consider the equation,

dvdt=g−cmv

Separate the variables as below,

dvg−cmv=dt

Integrate both the sides of above equation,

∫dvg−cmv=∫dt

Use the integration property of polynomial,

−ln(g−cmv)(cm)=t+C …… (1)

Where, C is the constant of integration.

For t=0, the initial velocity is v(0). Therefore,

−ln(g−cmv(0))(cm)=0+C−ln(g−cmv(0))(cm)=C

Substitute the value of C in the equation (1) and simplify as below,

−ln(g−cmv)(cm)=t−ln(g−cmv(0))(cm)−ln(g−cmv)=(cm)t−ln(g−cmv(0))−ln(g−cmv)+ln(g−cmv(0))=(cm)tln(g−cmv(0))(g−cmv)=(cm)t

Simplify furthermore,

(g−cmv(0))(g−cmv)=e(cm)tg−cmv=g−cmv(0)e(cm)tg−cmv=(g−cmv(0))e−(cm)t−cmv=(g−cmv(0))e−(cm)t−g

Simplify furthermore,

v=(−mc)(g−cmv(0))e−(cm)t−(−mc)g=−mgce−(cm)t+v(0)e−(cm)t+mgc=v(0)e−(cm)t+mgc(1−e−(cm)t)

Since, v(0) is non-zero. Hence, the solution of the equation is v=v(0)e−(cm)t+mgc(1−e−(cm)t).

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Students have asked these similar questions

Module Code: MATH380202 3. (a) Let {} be a white noise process with variance σ2. Define an ARMA(p,q) process {X} in terms of {+} and state (without proof) conditions for {X} to be (i) weakly stationary and (ii) invertible. Define what is meant by an ARIMA (p, d, q) process. Let {Y} be such an ARIMA(p, d, q) process and show how it can also be represented as an ARMA process, giving the AR and MA orders of this representation. (b) The following tables show the first nine sample autocorrelations and partial auto- correlations of X and Y₁ = VX+ for a series of n = 1095 observations. (Notice that the notation in this part has no relationship with the notation in part (a) of this question.) Identify a model for this time series and obtain preliminary estimates for the pa- rameters of your model. X₁ = 15.51, s² = 317.43. k 1 2 3 4 5 6 7 Pk 0.981 0.974 0.968 akk 0.981 0.327 8 9 0.927 0.963 0.957 0.951 0.943 0.935 0.121 0.104 0.000 0.014 -0.067 -0.068 -0.012 Y₁ = VX : y = 0.03, s² = 11.48. k 1…

Let G be a graph with n ≥ 2 vertices x1, x2, . . . , xn, and let A be the adjacency matrixof G. Prove that if G is connected, then every entry in the matrix A^n−1 + A^nis positive.

Module Code: MATH380202 1. (a) Define the terms "strongly stationary" and "weakly stationary". Let {X} be a stochastic process defined for all t € Z. Assuming that {X+} is weakly stationary, define the autocorrelation function (acf) Pk, for lag k. What conditions must a process {X+) satisfy for it to be white noise? (b) Let N(0, 1) for t€ Z, with the {+} being mutually independent. Which of the following processes {X+} are weakly stationary for t> 0? Briefly justify your answers. i. Xt for all > 0. ii. Xo~N(0,) and X₁ = 2X+-1+ &t for t > 0. (c) Provide an expression for estimating the autocovariance function for a sample X1,..., X believed to be from a weakly stationary process. How is the autocor- relation function Pk then estimated, and a correlogram (or acf plot) constructed? (d) Consider the weakly stationary stochastic process ✗+ = + + +-1+ +-2 where {E} is a white noise process with variance 1. Compute the population autocorre- lation function Pk for all k = 0, 1, ....

Answer 2

Textbook Question

Chapter 1, Problem 1P

Use calculus to solve Eq. (1.9) for the case where the initial velocity, v ( 0 ) is nonzero.

Expert Solution & Answer

To determine

To calculate: The solution of the equation dvdt=g−cmv by the use of calculus if the initial velocity v(0) is non-zero.

Answer to Problem 1P

Solution:

The solution of the equation dvdt=g−cmv is v=v(0)e−(cm)t+mgc(1−e−(cm)t).

Explanation of Solution

Given Information:

The equation, dvdt=g−cmv.

The initial velocity v(0) is non-zero.

Formula used:

Integration property of polynomial,

∫1a+bxdx=1bln(a+bx)+C

Logarithmic property,

lna−lnb=ln(ab)

Calculation:

Consider the equation,

dvdt=g−cmv

Separate the variables as below,

dvg−cmv=dt

Integrate both the sides of above equation,

∫dvg−cmv=∫dt

Use the integration property of polynomial,

−ln(g−cmv)(cm)=t+C …… (1)

Where, C is the constant of integration.

For t=0, the initial velocity is v(0). Therefore,

−ln(g−cmv(0))(cm)=0+C−ln(g−cmv(0))(cm)=C

Substitute the value of C in the equation (1) and simplify as below,

−ln(g−cmv)(cm)=t−ln(g−cmv(0))(cm)−ln(g−cmv)=(cm)t−ln(g−cmv(0))−ln(g−cmv)+ln(g−cmv(0))=(cm)tln(g−cmv(0))(g−cmv)=(cm)t

Simplify furthermore,

(g−cmv(0))(g−cmv)=e(cm)tg−cmv=g−cmv(0)e(cm)tg−cmv=(g−cmv(0))e−(cm)t−cmv=(g−cmv(0))e−(cm)t−g

Simplify furthermore,

v=(−mc)(g−cmv(0))e−(cm)t−(−mc)g=−mgce−(cm)t+v(0)e−(cm)t+mgc=v(0)e−(cm)t+mgc(1−e−(cm)t)

Since, v(0) is non-zero. Hence, the solution of the equation is v=v(0)e−(cm)t+mgc(1−e−(cm)t).

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 3

Textbook Question

Chapter 1, Problem 1P

Use calculus to solve Eq. (1.9) for the case where the initial velocity, v ( 0 ) is nonzero.

Expert Solution & Answer

To determine

To calculate: The solution of the equation dvdt=g−cmv by the use of calculus if the initial velocity v(0) is non-zero.

Answer to Problem 1P

Solution:

The solution of the equation dvdt=g−cmv is v=v(0)e−(cm)t+mgc(1−e−(cm)t).

Explanation of Solution

Given Information:

The equation, dvdt=g−cmv.

The initial velocity v(0) is non-zero.

Formula used:

Integration property of polynomial,

∫1a+bxdx=1bln(a+bx)+C

Logarithmic property,

lna−lnb=ln(ab)

Calculation:

Consider the equation,

dvdt=g−cmv

Separate the variables as below,

dvg−cmv=dt

Integrate both the sides of above equation,

∫dvg−cmv=∫dt

Use the integration property of polynomial,

−ln(g−cmv)(cm)=t+C …… (1)

Where, C is the constant of integration.

For t=0, the initial velocity is v(0). Therefore,

−ln(g−cmv(0))(cm)=0+C−ln(g−cmv(0))(cm)=C

Substitute the value of C in the equation (1) and simplify as below,

−ln(g−cmv)(cm)=t−ln(g−cmv(0))(cm)−ln(g−cmv)=(cm)t−ln(g−cmv(0))−ln(g−cmv)+ln(g−cmv(0))=(cm)tln(g−cmv(0))(g−cmv)=(cm)t

Simplify furthermore,

(g−cmv(0))(g−cmv)=e(cm)tg−cmv=g−cmv(0)e(cm)tg−cmv=(g−cmv(0))e−(cm)t−cmv=(g−cmv(0))e−(cm)t−g

Simplify furthermore,

v=(−mc)(g−cmv(0))e−(cm)t−(−mc)g=−mgce−(cm)t+v(0)e−(cm)t+mgc=v(0)e−(cm)t+mgc(1−e−(cm)t)

Since, v(0) is non-zero. Hence, the solution of the equation is v=v(0)e−(cm)t+mgc(1−e−(cm)t).

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Textbook Question

Chapter 1, Problem 1P

Use calculus to solve Eq. (1.9) for the case where the initial velocity, v ( 0 ) is nonzero.

Expert Solution & Answer

To determine

To calculate: The solution of the equation dvdt=g−cmv by the use of calculus if the initial velocity v(0) is non-zero.

Answer to Problem 1P

Solution:

The solution of the equation dvdt=g−cmv is v=v(0)e−(cm)t+mgc(1−e−(cm)t).

Explanation of Solution

Given Information:

The equation, dvdt=g−cmv.

The initial velocity v(0) is non-zero.

Formula used:

Integration property of polynomial,

∫1a+bxdx=1bln(a+bx)+C

Logarithmic property,

lna−lnb=ln(ab)

Calculation:

Consider the equation,

dvdt=g−cmv

Separate the variables as below,

dvg−cmv=dt

Integrate both the sides of above equation,

∫dvg−cmv=∫dt

Use the integration property of polynomial,

−ln(g−cmv)(cm)=t+C …… (1)

Where, C is the constant of integration.

For t=0, the initial velocity is v(0). Therefore,

−ln(g−cmv(0))(cm)=0+C−ln(g−cmv(0))(cm)=C

Substitute the value of C in the equation (1) and simplify as below,

−ln(g−cmv)(cm)=t−ln(g−cmv(0))(cm)−ln(g−cmv)=(cm)t−ln(g−cmv(0))−ln(g−cmv)+ln(g−cmv(0))=(cm)tln(g−cmv(0))(g−cmv)=(cm)t

Simplify furthermore,

(g−cmv(0))(g−cmv)=e(cm)tg−cmv=g−cmv(0)e(cm)tg−cmv=(g−cmv(0))e−(cm)t−cmv=(g−cmv(0))e−(cm)t−g

Simplify furthermore,

v=(−mc)(g−cmv(0))e−(cm)t−(−mc)g=−mgce−(cm)t+v(0)e−(cm)t+mgc=v(0)e−(cm)t+mgc(1−e−(cm)t)

Since, v(0) is non-zero. Hence, the solution of the equation is v=v(0)e−(cm)t+mgc(1−e−(cm)t).

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

Expert Solution & Answer

To determine

To calculate: The solution of the equation dvdt=g−cmv by the use of calculus if the initial velocity v(0) is non-zero.

Answer to Problem 1P

Solution:

The solution of the equation dvdt=g−cmv is v=v(0)e−(cm)t+mgc(1−e−(cm)t).

Explanation of Solution

Given Information:

The equation, dvdt=g−cmv.

The initial velocity v(0) is non-zero.

Formula used:

Integration property of polynomial,

∫1a+bxdx=1bln(a+bx)+C

Logarithmic property,

lna−lnb=ln(ab)

Calculation:

Consider the equation,

dvdt=g−cmv

Separate the variables as below,

dvg−cmv=dt

Integrate both the sides of above equation,

∫dvg−cmv=∫dt

Use the integration property of polynomial,

−ln(g−cmv)(cm)=t+C …… (1)

Where, C is the constant of integration.

For t=0, the initial velocity is v(0). Therefore,

−ln(g−cmv(0))(cm)=0+C−ln(g−cmv(0))(cm)=C

Substitute the value of C in the equation (1) and simplify as below,

−ln(g−cmv)(cm)=t−ln(g−cmv(0))(cm)−ln(g−cmv)=(cm)t−ln(g−cmv(0))−ln(g−cmv)+ln(g−cmv(0))=(cm)tln(g−cmv(0))(g−cmv)=(cm)t

Simplify furthermore,

(g−cmv(0))(g−cmv)=e(cm)tg−cmv=g−cmv(0)e(cm)tg−cmv=(g−cmv(0))e−(cm)t−cmv=(g−cmv(0))e−(cm)t−g

Simplify furthermore,

v=(−mc)(g−cmv(0))e−(cm)t−(−mc)g=−mgce−(cm)t+v(0)e−(cm)t+mgc=v(0)e−(cm)t+mgc(1−e−(cm)t)

Since, v(0) is non-zero. Hence, the solution of the equation is v=v(0)e−(cm)t+mgc(1−e−(cm)t).

Answer 8

Expert Solution & Answer

To determine

To calculate: The solution of the equation dvdt=g−cmv by the use of calculus if the initial velocity v(0) is non-zero.

Answer to Problem 1P

Solution:

The solution of the equation dvdt=g−cmv is v=v(0)e−(cm)t+mgc(1−e−(cm)t).

Explanation of Solution

Given Information:

The equation, dvdt=g−cmv.

The initial velocity v(0) is non-zero.

Formula used:

Integration property of polynomial,

∫1a+bxdx=1bln(a+bx)+C

Logarithmic property,

lna−lnb=ln(ab)

Calculation:

Consider the equation,

dvdt=g−cmv

Separate the variables as below,

dvg−cmv=dt

Integrate both the sides of above equation,

∫dvg−cmv=∫dt

Use the integration property of polynomial,

−ln(g−cmv)(cm)=t+C …… (1)

Where, C is the constant of integration.

For t=0, the initial velocity is v(0). Therefore,

−ln(g−cmv(0))(cm)=0+C−ln(g−cmv(0))(cm)=C

Substitute the value of C in the equation (1) and simplify as below,

−ln(g−cmv)(cm)=t−ln(g−cmv(0))(cm)−ln(g−cmv)=(cm)t−ln(g−cmv(0))−ln(g−cmv)+ln(g−cmv(0))=(cm)tln(g−cmv(0))(g−cmv)=(cm)t

Simplify furthermore,

(g−cmv(0))(g−cmv)=e(cm)tg−cmv=g−cmv(0)e(cm)tg−cmv=(g−cmv(0))e−(cm)t−cmv=(g−cmv(0))e−(cm)t−g

Simplify furthermore,

v=(−mc)(g−cmv(0))e−(cm)t−(−mc)g=−mgce−(cm)t+v(0)e−(cm)t+mgc=v(0)e−(cm)t+mgc(1−e−(cm)t)

Since, v(0) is non-zero. Hence, the solution of the equation is v=v(0)e−(cm)t+mgc(1−e−(cm)t).

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

To determine

To calculate: The solution of the equation dvdt=g−cmv by the use of calculus if the initial velocity v(0) is non-zero.

Answer 12

Answer to Problem 1P

Solution:

The solution of the equation dvdt=g−cmv is v=v(0)e−(cm)t+mgc(1−e−(cm)t).

Answer 13

Explanation of Solution

Given Information:

The equation, dvdt=g−cmv.

The initial velocity v(0) is non-zero.

Formula used:

Integration property of polynomial,

∫1a+bxdx=1bln(a+bx)+C

Logarithmic property,

lna−lnb=ln(ab)

Calculation:

Consider the equation,

dvdt=g−cmv

Separate the variables as below,

dvg−cmv=dt

Integrate both the sides of above equation,

∫dvg−cmv=∫dt

Use the integration property of polynomial,

−ln(g−cmv)(cm)=t+C …… (1)

Where, C is the constant of integration.

For t=0, the initial velocity is v(0). Therefore,

−ln(g−cmv(0))(cm)=0+C−ln(g−cmv(0))(cm)=C

Substitute the value of C in the equation (1) and simplify as below,

−ln(g−cmv)(cm)=t−ln(g−cmv(0))(cm)−ln(g−cmv)=(cm)t−ln(g−cmv(0))−ln(g−cmv)+ln(g−cmv(0))=(cm)tln(g−cmv(0))(g−cmv)=(cm)t

Simplify furthermore,

(g−cmv(0))(g−cmv)=e(cm)tg−cmv=g−cmv(0)e(cm)tg−cmv=(g−cmv(0))e−(cm)t−cmv=(g−cmv(0))e−(cm)t−g

Simplify furthermore,

v=(−mc)(g−cmv(0))e−(cm)t−(−mc)g=−mgce−(cm)t+v(0)e−(cm)t+mgc=v(0)e−(cm)t+mgc(1−e−(cm)t)

Since, v(0) is non-zero. Hence, the solution of the equation is v=v(0)e−(cm)t+mgc(1−e−(cm)t).

Answer 14

Ch. 1 - Use calculus to solve Eq. (1.9) for the case where...Ch. 1 - 1.2 Repeat Example 1.2. Compute the velocity to ...Ch. 1 - Rather than the linear relationship of Eq. (1.7),...Ch. 1 - For the free-falling parachutist with linear drag,...Ch. 1 - Compute the velocity of a free-falling parachutist...Ch. 1 - 1.6 The following information is available for a...Ch. 1 - The amount of a uniformly distributed radioactive...Ch. 1 - A group of 35 students attend a class in a room...Ch. 1 - A storage tank contains a liquid at depth y, where...Ch. 1 - For the same storage tank described in Prob. 1.9,...

Use calculus to solve Eq. (1.9) for the case where the initial velocity, v ( 0 ) is nonzero.

Concept explainers

Videos

Answer to Problem 1P

Explanation of Solution

Want to see more full solutions like this?

Chapter 1 Solutions