Motion problems

Motion Problems are all about this relationships: Moving position -> Velocity(or speed) -> Acceleration.

These terms are constantly confusing people, especially the follow parts:

• Velocity is NOT the derivative of speed, but only the speed with a direction: s(t) = |v(t)|.

• Velocity IS the derivative of Position: v(t) = p'(t)

• Acceleration is the derivative of the Velocity: a(t) = v'(t)

• Max or Min Position means Velocity = v(t) = p'(t) = 0

• Max or Min Velocity means Acceleration = a(t) = v'(t) = 0

• Max or Min Acceleration means a'(t) = v''(t) = p'''(t) = 0

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Example

Solve:

• The tricky part here is the relationships: Position -> Velocity -> Acceleration

  • Position: p(t) = x(t)

  • Velocity: v(t) = x'(t)

  • Acceleration: a(t) = v'(t) = x''(t)

• To conclude, the Max velocity should satisfy this: a(t) = 0 & a'(t) < 0

• Differentiate x(t) twice and set x''(t) = 0, get t = 1.

Example

Solve:

• The velocity is v(t) = x'(t)

• The Acceleration is a(t) = v'(t) = x''(t) = 0, and get t=1

• Substitute to v(1) = 3

Example

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►Refer to the note: Related rates.