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yanalaym [24]
3 years ago
6

The acceleration of an object is =4 m/s^2.?

Mathematics
1 answer:
Pie3 years ago
6 0

Answer:

\displaystyle s(t) = \frac{2t^3}{3} + 16t + 9

General Formulas and Concepts:

<u>Pre-Algebra</u>

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right  

<u>Calculus</u>

Antiderivatives - Integrals

Integration Constant C

Integration Rule [Reverse Power Rule]:                                                               \displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C

Integration Property [Multiplied Constant]:                                                         \displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx

Integration Property [Addition/Subtraction]:                                                       \displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx

Step-by-step explanation:

*Note:

Velocity is the derivative of Position, and Acceleration is derivative of Velocity.

↓

Velocity is integration of Acceleration, Position is integration of Velocity.

<u>Step 1: Define</u>

a(t) = 4t m/s²

s(0) = 9 m

v(0) = 16 m/s

<u>Step 2: Find Velocity Function</u>

<em>Integration Pt. 1</em>

  1. [Velocity] Set up integral:                                                                             \displaystyle v(t) = \int {a(t)} \, dt
  2. [Velocity] Substitute in function:                                                                   \displaystyle v(t) = \int {4t} \, dt
  3. [Velocity] Rewrite [Integration Property - Multiplied Constant]:                 \displaystyle v(t) = 4\int {t} \, dt
  4. [Velocity] Integrate [Integration Rule - Reverse Power Rule]:                   \displaystyle v(t) = 4(\frac{t^2}{2}) + C
  5. [Velocity] Multiply:                                                                                         \displaystyle v(t) = 2t^2 + C

<em>Finding C</em>

  1. [Velocity] Substitute in initial condition:                                                       \displaystyle v(0) = 2(0)^2 + C
  2. [Velocity] Substitute in function value:                                                         \displaystyle 16 = 2(0)^2 + C
  3. [Velocity] Evaluate exponents:                                                                     \displaystyle 16 = 2(0) + C
  4. [Velocity] Multiply:                                                                                         \displaystyle 16 = C
  5. Rewrite:                                                                                                         \displaystyle C = 16

Velocity Function: \displaystyle v(t) = 2t^2 + 16

<u>Step 3: Find Position Function</u>

<em>Integration Pt. 2</em>

  1. [Position] Set up integral:                                                                             \displaystyle s(t) = \int {v(t)} \, dt
  2. [Position] Substitute in function:                                                                   \displaystyle s(t) = \int {2t^2 + 16} \, dt
  3. [Position] Rewrite [Integration Property - Addition]:                                    \displaystyle s(t) = \int {2t^2} \, dt + \int {16} \, dt
  4. [Position] Rewrite [Integration Property - Multiplied Constant]:                 \displaystyle s(t) = 2\int {t^2} \, dt + 16\int {} \, dt
  5. [Position] Integrate [Integration Rule - Reverse Power Rule]:                   \displaystyle s(t) = 2(\frac{t^3}{3}) + 16t + C
  6. [Position] Multiply:                                                                                         \displaystyle s(t) = \frac{2t^3}{3} + 16t + C

<em>Finding C</em>

  1. [Position] Substitute in initial condition:                                                       \displaystyle s(0) = \frac{2(0)^3}{3} + 16(0) + C
  2. [Position] Substitute in function value:                                                         \displaystyle 9 = \frac{2(0)^3}{3} + 16(0) + C
  3. [Position] Evaluate exponents:                                                                     \displaystyle 9 = \frac{2(0)}{3} + 16(0) + C
  4. [Position] Multiply:                                                                                         \displaystyle 9 = \frac{0}{3} + C
  5. [Position] Divide:                                                                                           \displaystyle 9 = C
  6. [Position] Rewrite:                                                                                         \displaystyle C = 9

Position Function: \displaystyle s(t) = \frac{2t^3}{3} + 16t + 9

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Integration

Book: College Calculus 10e

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