DISPLACEMENT vs. DISTANCE TRAVELED
If a body with position function s (t) moves along a coordinate line without changing direction,
we can calculate the total distance it travels from t = a to t = b. If the body changes direction one
or more times during the trip, then we need to integrate the body's speed |v (t)| to find the total
distance traveled.
FACT:
FACT:
EXAMPLE 1:
Find the total distance traveled by a body and the body's
displacement for a body whose velocity is v (t) = 6sin 3t on the
time interval 0 t /2.
SOLUTION: To find the distance traveled, we need to find the values of t where the function
changes direction. To do this, set v (t) = 0 and solve for t.
t = 0 is the starting point, and t = 2 /3 is not in the time interval. Now to determine which
direction the body is going on the time intervals (0, /3) and ( /3, /2).
(0, /3)
v ( /4) > 0
( /3, /2)
v (5 /12) < 0
So to find the total distance traveled, I will have two integrals. One will have to be taken times a
-1 to make it positive.
Displacement will be the integral from 0 to /2.
