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) = 0Max or Min Velocity means
Acceleration = a(t) = v'(t) = 0Max or Min Acceleration means
a'(t) = v''(t) = p'''(t) = 0
Jump over here for Khan academy's quizzes.
Example
The tricky part here is the relationships:
Position -> Velocity -> AccelerationPosition:
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) < 0Differentiate
x(t)twice and setx''(t) = 0, gett = 1.
Example
The velocity is
v(t) = x'(t)The Acceleration is
a(t) = v'(t) = x''(t) = 0, and gett=1Substitute to
v(1) = 3
Example
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