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An object has a constant acceleration of 10 ft/sec2, an initial velocity of -20 ft/sec, and an initial position of 8 ft. Find the position function, s(t), describing the motion of the object.

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Answer:

[tex]\displaystyle s(t) = 5t^2 - 20t + 8[/tex]

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

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

Algebra I

Functions

  • Function Notation

Calculus

Differentiation

  • Derivatives
  • Derivative Notation
  • Position/Velocity/Acceleration

Integration

  • Integrals
  • Indefinite Integrals
  • Integration Constant C
  • Position/Velocity/Acceleration

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

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

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

Step-by-step explanation:

Step 1: Define

Identify

a(t) = 10 ft/sec²

v(0) = -20 ft/sec

s(0) = 8 ft

Step 2: Find Velocity Function

  1. Define:                                                                                                           [tex]\displaystyle v(t) = \int {a(t)} \, dt[/tex]
  2. [Integrand] Substitute in acceleration function:                                         [tex]\displaystyle v(t) = \int {10} \, dt[/tex]
  3. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle v(t) = 10\int {} \, dt[/tex]
  4. [Integral] Integrate [Integration Rule - Reverse Power Rule]:                     [tex]\displaystyle v(t) = 10t + C[/tex]
  5. Substitute in t [Velocity Function v(t)]:                                                         [tex]\displaystyle v(0) = 10(0) + C[/tex]
  6. Substitute in function values:                                                                       [tex]\displaystyle -20 = 10(0) + C[/tex]
  7. Solve:                                                                                                             [tex]\displaystyle C = -20[/tex]
  8. Substitute in C [Velocity Function v(t)]:                                                       [tex]\displaystyle v(t) = 10t - 20[/tex]

Step 3: Find Position Function

  1. Define:                                                                                                           [tex]\displaystyle s(t) = \int {v(t)} \, dt[/tex]
  2. [Integrand] Substitute in velocity function:                                                 [tex]\displaystyle s(t) = \int {(10t - 20)} \, dt[/tex]
  3. [Integral] Rewrite [Integration Property - Addition/Subtraction]:               [tex]\displaystyle s(t) = \int {10t} \, dt - \int {20} \, dt[/tex]
  4. [Integrals] Rewrite [Integration Property - Multiplied Constant]:               [tex]\displaystyle s(t) = 10\int {t} \, dt - 20\int {} \, dt[/tex]
  5. [Integrals] Integrate [Integration Rule - Reverse Power Rule]:                   [tex]\displaystyle s(t) = 10(\frac{t^2}{2}) - 20t + C[/tex]
  6. Simplify:                                                                                                         [tex]\displaystyle s(t) = 5t^2 - 20t + C[/tex]
  7. Substitute in t [Position Function s(t)]:                                                         [tex]\displaystyle s(0) = 5(0)^2 - 20(0) + C[/tex]
  8. Substitute in function values:                                                                       [tex]\displaystyle 8 = 5(0)^2 - 20(0) + C[/tex]
  9. Solve:                                                                                                             [tex]\displaystyle C = 8[/tex]
  10. Substitute in C [Position Function s(t)]:                                                         [tex]\displaystyle s(t) = 5t^2 - 20t + 8[/tex]

∴ our position function is  [tex]\displaystyle s(t) = 5t^2 - 20t + 8[/tex].

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

Unit: Integration

Book: College Calculus 10e