Calculus I (AB) and Calculus II (BC): Distance Traveled
If a particle (moving left and right along the x-axis) has velocity equation
, and its initial position is 7, find:
(a) The position equation, s(t)
(b) The total distance traveled by the particle on the time interval [0,5]
Difficulty:
Solution:(a) Since position is the antiderivative of velocity, integrate v(t) to find s(t):
However, this answer is not as accurate as it can be. We're told in the problem that s(0) = 7, so plug in 0 for t and 7 for s(t) to find C:
Therefore,
.
(b) Before you can calculate distance traveled, you have to see if the particle changes direction on the given interval and integrate velocity separately on each interval of time separated by this change of direction. If you set velocity equal to 0, you get
Only the 3 falls within the interval we're concerned about, and velocity does, indeed, change signs around this value of t (verify with a wiggle graph if you like). So, we have to integrate velocity twice, once on the interval [0,3] and once on the interval [3,5]. Note: you don't have to see if the velocity changes signs at 3; you'll still get the correct answer if you split up the integral, even if it is unnecessary.
This tells us that the particle travels backwards 27/2 units and then forwards 38/3 units, or
units in all.
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