Chapter 3: Q11E (page 116)

In Example 1, we solved for the velocity of the object as a function of time (equation (5)). In some cases, it is useful to have an expression, independent of t, that relates vand x.Find this relation for the motion in Example 1. [Hint: Letting $v (t) = V (x (t))$ , then $\frac{d v}{d t} = (\frac{d v}{d x}) V$ ]

Short Answer

Expert verified

The relation for the motion is $e^{b V} {|m g - b V|}^{m g} = e^{b v_{0}} {|m g - b v_{0}|}^{m g} e^{\frac{- b^{2} x}{m}}$ .

Step by step solution

Important hints.

To solve this question, use chain rule, and integration rules.

Find the equation of motion for this relation

Here given is $v (t) = V (x (t))$

Apply chain rule for the solution $\frac{d v}{d t} = \frac{d v}{d x} . \frac{d x}{d t}$

$v \frac{d v}{d x} = V \frac{d x}{d t}$

Now

$m V \frac{d v}{d t} = m g - b v \frac{m V d V}{m g - b V} = d x (Variableseparating) \frac{m V d V}{m g - b V} = d x m \int \frac{V d v}{m g - b V} = x + C$

Finding the value of ∫Vdvmg-bV

Now,

$m \int \frac{V d v}{m g - b V} = |m g - b V = t d V = - \frac{d t}{b}| (V = \frac{m g - t}{b})$

$\frac{1}{b^{2}} \int \frac{t - m g}{t} d t \frac{1}{b^{2}} \int d t - m g \int \frac{d t}{t} \frac{1}{b^{2}} (t - m g I n | t |) + c_{1} \frac{1}{b^{2}} (m g - b V - m g I n | m g - b V |) + c_{1} \frac{m g}{b^{2}} - \frac{V}{b} - \frac{m g \ln |m g - b V|}{b^{2}} + c_{1}$

Now using then $C - \frac{m g}{b^{2}} = A$ then

$- \frac{V}{b} - \frac{g m \ln |m g - b V|}{b^{2}} + D = \frac{x}{m} + A - b V - g m \ln |m g - b V| = \frac{b^{2} x}{m} + B g m \ln |m g - b V| = - \frac{b^{2} x}{m} - b V - B \ln {|m g - b V|}^{g m} = - \frac{b^{2} x}{m} - b V - B {|m g - b V|}^{g m} = P e^{\frac{- b^{2} x}{m}} e^{- b V} e^{b V} {|m g - b V|}^{g m} = P e^{\frac{- b^{2} x}{m}}$

When then

V (0) = v_{0}

then

e^{b v_{0}} {|m g - b v_{0}|}^{m g} = P e^{0}

Therefore, role="math" localid="1663931337161" $e^{bV} {|mg - bV|}^{mg} = e^{{bv}_{0}} {|mg - {bv}_{0}|}^{mg} e^{\frac{- b^{2} x}{m}}$

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Short Answer

Step by step solution

Important hints.

Find the equation of motion for this relation

Finding the value of ∫Vdvmg-bV

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In Example 1, we solved for the velocity of the object as a function of time (equation (5)). In some cases, it is useful to have an expression, independent of t, that relates vand x.Find this relation for the motion in Example 1. [Hint: Lettingv(t)=V(xt), thendvdt=(dvdx)V]

Short Answer

Step by step solution

Important hints.

Find the equation of motion for this relation

Finding the value of ∫Vdvmg-bV

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Statistics

Pure Maths

Logic and Functions

Mechanics Maths

Decision Maths

Study anywhere. Anytime. Across all devices.