88
Chapter 3Acceleration and Free Fall
That’s what I’ve done above. Each rectangle on the graph paper is 1.0 s
wide and 2 m/s tall, so it represents 2 m. Adding up all the numbers gives
.
x=41 m. If you needed better accuracy, you could use graph paper with
smaller rectangles.
It’s important to realize that this technique gives you
.
x, not x. The v-t
graph has no information about where the object was when it started.
The following are important points to keep in mind when applying this
technique:
•
If the range of v values on your graph does not extend down to
zero, then you will get the wrong answer unless you compensate by
adding in the area that is not shown.
•
As in the example, one rectangle on the graph paper does not
necessarily correspond to one meter of distance.
•
Negative velocity values represent motion in the opposite direction,
so area under the t axis should be subtracted, i.e. counted as
“negative area.”
•
Since the result is a
.
x value, it only tells you x
after
-x
before
, which
may be less than the actual distance traveled. For instance, the
object could come back to its original position at the end, which
would correspond to
.
x=0, even though it had actually moved a
nonzero distance.
Finally, note that one can find
.
v from an a-t graph using an entirely
analogous method. Each rectangle on the a-t graph represents a certain
amount of velocity change.
Discussion question
Roughly what would a pendulum’s v-t graph look like. What would happen
when you applied the area-under-the-curve technique to find the pendulum’s
.
x for a time period covering many swings.
0
10
20
02468
t (s)
2 m
1 m
2 m
2 m
2 m
0.5 m
2 m
2 m
2 m
2 m
2 m
2 m
2 m
1.5 m
1 m
1 m
1.5 m
1.5 m
1.5 m
1.5 m
1.5 m
1.5 m
1.5 m
1.5 m
1.5 m
1.5 m
0.5 m