Resource Lesson
Derivatives: Instantaneous vs Average Velocities
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As we have already discussed, there are distinctions that need to be remembered when using the terms
distance traveled, net displacement, average speed, average velocity, instantaneous speed, and instantaneous velocity
. There are times when certain terms are synonymous and other times when they are absolutely and completely different.
A few examples to refresh your memories:
While traveling in one direction along a straight line path,
total distance traveled
and
net displacement
would be equivalent HOWEVER, if the path changed direction at any time, those two terms would then be completely different with distance representing the length of the path traveled and displacement representing the length of the straight-line vector that begins at the starting point and ends at the finishing point, independent of the path taken.
In both of these examples, the displacements are identical whereas the distance traveled between points C and D is much greater than the distance traveled between points A and B since the direction switches back and forth causing the path length to be much longer.
While traveling at a
constant speed
, the calculations of an object's constant speed,
average speed, and instantaneous speed
would all yield equivalent values HOWEVER, if the speed was varied throughout the internal then the average speed during that interval would only occasionally equal its instantaneous speed at any one moment in the interval. Obviously, the notion of a constant speed would no longer be discussed under those conditions.
Both of these graph represent position vs time graphs for objects traveling along straight-line paths. On the left, the first object travels at a constant speed beginning at x = 5 and ending at x = -6 taking approximately 5.5 seconds to complete the trip. On the right graph, a second object is traveling at a variable rate beginning at x = 1 traveling down to x = 0 then up to x = 6 and finally all the way back to x = -8 taking approximately 5 seconds to complete its trip.
Average speed and average velocity
during an interval also take on their own unique properties. Average speed represents the ratio of total distance traveled divided by the total time taken to travel that distance while average velocity represents the ratio of the object's net displacement divided by the total time required to achieve that displacement. Whenever total distance traveled and net displacement differ, average speed and average velocity will be different. Since average speed is a scalar it is always reported as a positive (or zero) value; while average velocity (being a vector quantity) can have a negative, zero, or positive value depending on how you assign positive/negative directions in setting up the problem.
On the left graph displayed above the first object's average speed (11 units/ 5.5 seconds) equals the magnitude of its average velocity (11 units / 5.5 seconds, down). While on the right graph the second object's average velocity (9 units / 5 seconds, down) is much smaller than its average speed (15 units / 5 seconds).
On the other hand, an object's
instantaneous speed
will always equal the
magnitude of its instantaneous velocity
.
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