AP Calculus A/B: Particle Motion

Average velocity on [a,b]

(1/(b-a))∫v(t)dt (where the integral limits are from b to a)

Total distance traveled on [a,b]

Integral |v(t)|dt *absolute value is important*

Acceleration at time t=c

a(c)=v(v)'=x"(c)

Velocity is decreasing

a(t)=v(t)'<0

Velocity is increasing

a(t)=v(t)'>0

Particle is farthest to the left (right)

compare positions (x-values) at endpoints and at local minima (maxima)

Initially

t=0

Instantaneous Velocity at time t=c

v(c)=x'(c)

Speed is decreasing (slowing down)

v(t) and a(t) have different signs

Speed is increasing (speeding up)

v(t) and a(t) have same sign (both are positive or negative)

Particle moving left (backward or down)

v(t)<0

At rest

v(t)=0

Particle moving right (forward or up)

v(t)>0

Position of object at time

x(b)=x(a)+∫v(t)dt

Speed

|v(t)|

Net distance traveled (displacement)

∫v(t)dt