Calculus Examples

$\frac{d v}{d t} = 9.8 - 0.196 v$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{9.8 - 0.196 v}$ .

$\frac{1}{9.8 - 0.196 v} \frac{d v}{d t} = \frac{1}{9.8 - 0.196 v} (9.8 - 0.196 v)$

Step 1.2

Cancel the common factor of $9.8 - 0.196 v$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{9.8 - 0.196 v} \frac{d v}{d t} = \frac{1}{9.8 - 0.196 v} (9.8 - 0.196 v)$

Step 1.2.2

Rewrite the expression.

$\frac{1}{9.8 - 0.196 v} \frac{d v}{d t} = 1$

Step 1.3

Rewrite the equation.

$\frac{1}{9.8 - 0.196 v} d v = 1 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{9.8 - 0.196 v} d v = \int d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = 9.8 - 0.196 v$ . Then $d u = - 0.196 d v$ , so $\frac{1}{- 0.196} d u = d v$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = 9.8 - 0.196 v$ . Find $\frac{d u}{d v}$ .

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Step 2.2.1.1.1

Differentiate $9.8 - 0.196 v$ .

$\frac{d}{d v} [9.8 - 0.196 v]$

Step 2.2.1.1.2

Differentiate.

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Step 2.2.1.1.2.1

By the Sum Rule, the derivative of $9.8 - 0.196 v$ with respect to $v$ is $\frac{d}{d v} [9.8] + \frac{d}{d v} [- 0.196 v]$ .

$\frac{d}{d v} [9.8] + \frac{d}{d v} [- 0.196 v]$

Step 2.2.1.1.2.2

Since $9.8$ is constant with respect to $v$ , the derivative of $9.8$ with respect to $v$ is $0$ .

$0 + \frac{d}{d v} [- 0.196 v]$

Step 2.2.1.1.3

Evaluate $\frac{d}{d v} [- 0.196 v]$ .

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Step 2.2.1.1.3.1

Since $- 0.196$ is constant with respect to $v$ , the derivative of $- 0.196 v$ with respect to $v$ is $- 0.196 \frac{d}{d v} [v]$ .

$0 - 0.196 \frac{d}{d v} [v]$

Step 2.2.1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d v} [v^{n}]$ is $n v^{n - 1}$ where $n = 1$ .

$0 - 0.196 \cdot 1$

Step 2.2.1.1.3.3

Multiply $- 0.196$ by $1$ .

$0 - 0.196$

Step 2.2.1.1.4

Subtract $0.196$ from $0$ .

$- 0.196$

Step 2.2.1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{u} \cdot \frac{1}{- 0.196} d u = \int d t$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Move the negative in front of the fraction.

$\int \frac{1}{u} (- \frac{1}{0.196}) d u = \int d t$

Step 2.2.2.2

Multiply $\frac{1}{u}$ by $\frac{1}{0.196}$ .

$\int - \frac{1}{u \cdot 0.196} d u = \int d t$

Step 2.2.2.3

Move $0.196$ to the left of $u$ .

$\int - \frac{1}{0.196 u} d u = \int d t$

Step 2.2.3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int \frac{1}{0.196 u} d u = \int d t$

Step 2.2.4

Since $\frac{1}{0.196}$ is constant with respect to $u$ , move $\frac{1}{0.196}$ out of the integral.

$- (\frac{1}{0.196} \int \frac{1}{u} d u) = \int d t$

Step 2.2.5

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$- \frac{1}{0.196} (\ln (| u |) + C_{1}) = \int d t$

Step 2.2.6

Simplify.

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Step 2.2.6.1

Rewrite $- \frac{1}{0.196} (\ln (| u |) + C_{1})$ as $- 1 \cdot 5.10204081 \ln (| u |) + C_{1}$ .

$- 1 \cdot 5.10204081 \ln (| u |) + C_{1} = \int d t$

Step 2.2.6.2

Multiply $- 1$ by $5.10204081$ .

$- 5.10204081 \ln (| u |) + C_{1} = \int d t$

Step 2.2.7

Replace all occurrences of $u$ with $9.8 - 0.196 v$ .

$- 5.10204081 \ln (| 9.8 - 0.196 v |) + C_{1} = \int d t$

Step 2.3

Apply the constant rule.

$- 5.10204081 \ln (| 9.8 - 0.196 v |) + C_{1} = t + C_{2}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$- 5.10204081 \ln (| 9.8 - 0.196 v |) = t + C$

Step 3

Solve for $v$ .

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Step 3.1

Divide each term in $- 5.10204081 \ln (| 9.8 - 0.196 v |) = t + C$ by $- 5.10204081$ and simplify.

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Step 3.1.1

Divide each term in $- 5.10204081 \ln (| 9.8 - 0.196 v |) = t + C$ by $- 5.10204081$ .

$\frac{- 5.10204081 \ln (| 9.8 - 0.196 v |)}{- 5.10204081} = \frac{t}{- 5.10204081} + \frac{C}{- 5.10204081}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $- 5.10204081$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{- 5.10204081 \ln (| 9.8 - 0.196 v |)}{- 5.10204081} = \frac{t}{- 5.10204081} + \frac{C}{- 5.10204081}$

Step 3.1.2.1.2

Divide $\ln (| 9.8 - 0.196 v |)$ by $1$ .

$\ln (| 9.8 - 0.196 v |) = \frac{t}{- 5.10204081} + \frac{C}{- 5.10204081}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Move the negative in front of the fraction.

$\ln (| 9.8 - 0.196 v |) = - \frac{t}{5.10204081} + \frac{C}{- 5.10204081}$

Step 3.1.3.1.2

Multiply by $1$ .

$\ln (| 9.8 - 0.196 v |) = - \frac{1 t}{5.10204081} + \frac{C}{- 5.10204081}$

Step 3.1.3.1.3

Factor $5.10204081$ out of $5.10204081$ .

$\ln (| 9.8 - 0.196 v |) = - \frac{1 t}{5.10204081 (1)} + \frac{C}{- 5.10204081}$

Step 3.1.3.1.4

Separate fractions.

$\ln (| 9.8 - 0.196 v |) = - (\frac{1}{5.10204081} \cdot \frac{t}{1}) + \frac{C}{- 5.10204081}$

Step 3.1.3.1.5

Divide $1$ by $5.10204081$ .

$\ln (| 9.8 - 0.196 v |) = - (0.196 \frac{t}{1}) + \frac{C}{- 5.10204081}$

Step 3.1.3.1.6

Divide $t$ by $1$ .

$\ln (| 9.8 - 0.196 v |) = - (0.196 t) + \frac{C}{- 5.10204081}$

Step 3.1.3.1.7

Multiply $0.196$ by $- 1$ .

$\ln (| 9.8 - 0.196 v |) = - 0.196 t + \frac{C}{- 5.10204081}$

Step 3.1.3.1.8

Move the negative in front of the fraction.

$\ln (| 9.8 - 0.196 v |) = - 0.196 t - \frac{C}{5.10204081}$

Step 3.1.3.1.9

Multiply by $1$ .

$\ln (| 9.8 - 0.196 v |) = - 0.196 t - \frac{1 C}{5.10204081}$

Step 3.1.3.1.10

Factor $5.10204081$ out of $5.10204081$ .

$\ln (| 9.8 - 0.196 v |) = - 0.196 t - \frac{1 C}{5.10204081 (1)}$

Step 3.1.3.1.11

Separate fractions.

$\ln (| 9.8 - 0.196 v |) = - 0.196 t - (\frac{1}{5.10204081} \cdot \frac{C}{1})$

Step 3.1.3.1.12

Divide $1$ by $5.10204081$ .

$\ln (| 9.8 - 0.196 v |) = - 0.196 t - (0.196 \frac{C}{1})$

Step 3.1.3.1.13

Divide $C$ by $1$ .

$\ln (| 9.8 - 0.196 v |) = - 0.196 t - (0.196 C)$

Step 3.1.3.1.14

Multiply $0.196$ by $- 1$ .

$\ln (| 9.8 - 0.196 v |) = - 0.196 t - 0.196 C$

Step 3.2

To solve for $v$ , rewrite the equation using properties of logarithms.

$e^{\ln (| 9.8 - 0.196 v |)} = e^{- 0.196 t - 0.196 C}$

Step 3.3

Rewrite $\ln (| 9.8 - 0.196 v |) = - 0.196 t - 0.196 C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{- 0.196 t - 0.196 C} = | 9.8 - 0.196 v |$

Step 3.4

Solve for $v$ .

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Step 3.4.1

Rewrite the equation as $| 9.8 - 0.196 v | = e^{- 0.196 t - 0.196 C}$ .

$| 9.8 - 0.196 v | = e^{- 0.196 t - 0.196 C}$

Step 3.4.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$9.8 - 0.196 v = \pm e^{- 0.196 t - 0.196 C}$

Step 3.4.3

Subtract $9.8$ from both sides of the equation.

$- 0.196 v = \pm e^{- 0.196 t - 0.196 C} - 9.8$

Step 3.4.4

Divide each term in $- 0.196 v = \pm e^{- 0.196 t - 0.196 C} - 9.8$ by $- 0.196$ and simplify.

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Step 3.4.4.1

Divide each term in $- 0.196 v = \pm e^{- 0.196 t - 0.196 C} - 9.8$ by $- 0.196$ .

$\frac{- 0.196 v}{- 0.196} = \frac{\pm e^{- 0.196 t - 0.196 C}}{- 0.196} + \frac{- 9.8}{- 0.196}$

Step 3.4.4.2

Simplify the left side.

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Step 3.4.4.2.1

Cancel the common factor of $- 0.196$ .

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Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{- 0.196 v}{- 0.196} = \frac{\pm e^{- 0.196 t - 0.196 C}}{- 0.196} + \frac{- 9.8}{- 0.196}$

Step 3.4.4.2.1.2

Divide $v$ by $1$ .

$v = \frac{\pm e^{- 0.196 t - 0.196 C}}{- 0.196} + \frac{- 9.8}{- 0.196}$

Step 3.4.4.3

Simplify the right side.

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Step 3.4.4.3.1

Simplify each term.

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Step 3.4.4.3.1.1

Simplify $\frac{\pm e^{- 0.196 t - 0.196 C}}{- 0.196}$ .

$v = \frac{\pm e^{- 0.196 t - 0.196 C}}{0.196} + \frac{- 9.8}{- 0.196}$

Step 3.4.4.3.1.2

Multiply by $1$ .

$v = \frac{1 (\pm e^{- 0.196 t - 0.196 C})}{0.196} + \frac{- 9.8}{- 0.196}$

Step 3.4.4.3.1.3

Factor $0.196$ out of $0.196$ .

$v = \frac{1 (\pm e^{- 0.196 t - 0.196 C})}{0.196 (1)} + \frac{- 9.8}{- 0.196}$

Step 3.4.4.3.1.4

Separate fractions.

$v = \frac{1}{0.196} \cdot \frac{\pm e^{- 0.196 t - 0.196 C}}{1} + \frac{- 9.8}{- 0.196}$

Step 3.4.4.3.1.5

Divide $1$ by $0.196$ .

$v = 5.10204081 \frac{\pm e^{- 0.196 t - 0.196 C}}{1} + \frac{- 9.8}{- 0.196}$

Step 3.4.4.3.1.6

Divide $\pm e^{- 0.196 t - 0.196 C}$ by $1$ .

$v = 5.10204081 (\pm e^{- 0.196 t - 0.196 C}) + \frac{- 9.8}{- 0.196}$

Step 3.4.4.3.1.7

Divide $- 9.8$ by $- 0.196$ .

$v = 5.10204081 (\pm e^{- 0.196 t - 0.196 C}) + 50$

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

$v = 5.10204081 (\pm e^{- 0.196 t + C}) + 50$

Step 4.2

Rewrite $e^{- 0.196 t + C}$ as $e^{- 0.196 t} e^{C}$ .

$v = 5.10204081 (\pm e^{- 0.196 t} e^{C}) + 50$

Step 4.3

Reorder $e^{- 0.196 t}$ and $e^{C}$ .

$v = 5.10204081 (\pm e^{C} e^{- 0.196 t}) + 50$

Step 4.4

Combine constants with the plus or minus.

$v = 5.10204081 (C e^{- 0.196 t}) + 50$