A particle moves along a straight line with the acceleration a = (12t - 3t ^ 1/2) feet / s ^ 2, where t is in seconds. Determine your speed and position as a function of time. When t = 0, v = 0 and s = 15 feet.

v = 6t² − 2t^³/₂

s = 2t³ − ⅘t^⁵/₂ + 15

Explanation:

a = 12t − 3t^½

Integrate to find velocity.

v = ∫ a dt

v = ∫ (12t − 3t^½) dt

v = 6t² − 2t^³/₂ + C

Use initial condition to find C.

0 = 6(0)² − 2(0)^³/₂ + C

C = 0

v = 6t² − 2t^³/₂

Integrate to find position.

s = ∫ v dt

s = ∫ (6t² − 2t^³/₂) dt

s = 2t³ − ⅘t^⁵/₂ + C

Use initial condition to find C.

15 = 2(0)³ − ⅘(0)^⁵/₂ + C

15 = C

s = 2t³ − ⅘t^⁵/₂ + 15

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