course Phy 231
A graph is constructed representing velocity vs. clock time for the interval between clock times t = 5 seconds and t = 13 seconds. The graph consists of a straight line from the point (5 sec, 16 cm/s) to the point (13 sec, 40 cm/s). What is the clock time at the midpoint of this interval?
answer/question/discussion:
midpoint clocktime = (13 s - 5 s) / 2 + 5 s = 9 s
What is the velocity at the midpoint of this interval?
answer/question/discussion:
midpt vel = (40 cm/s - 16 cm/s) / 2 + 16 cm/s
midpt vel = 28 cm/s
How far do you think the object travels during this interval?
answer/question/discussion:
between the 5s mark and the 9 s mark, it travels:
'ds = vAve * 'dt = 28 cm/s * 4s = 112 cm
if we're talking between the 5 s and 13 s mark:
28 cm/s * 8 s = 224 cm/s
By how much does the clock time change during this interval?
answer/question/discussion:
9 s - 5s = 4 s
if we're talking the entire interval:
13 s - 5 s = 8 s
By how much does velocity change during this interval?
answer/question/discussion:
'dv = vf - v0 = 28 cm/s - 16 cm/s = 12 cm/s
or if we're talking about the entire interval:
'dv = 40 cm/s - 16 cm/s = 24 cm/s
What is the average rate of change of velocity with respect to clock time on this interval?
answer/question/discussion:
a = 'dv / 'dt = 12 cm/s / 4 s = 3 cm/s^2
OR
a = 24 cm/s / 8 s = 3 cm/s^2
What is the rise of the graph between these points?
answer/question/discussion:
28 cm/s - 16 cm/s = 12 cm/s
OR
40 cm/s - 16 cm/s = 24 cm/s
What is the run of the graph between these points?
answer/question/discussion:
9 s - 5 s = 4 s
OR
13 s - 5 s = 8 s
What is the slope of the graph between these points?
answer/question/discussion:
slope = rise/run = 12 cm/s / 4 s = 3 cm/s^2
OR
24 cm/s / 8 s = 3 cm/s^2
What does the slope of the graph tell you about the motion of the object during this interval?
answer/question/discussion:
the slope of the graph of velocity vs clock time is equal to the acceleration of the object
The appropriate solution uses the given interval, as opposed to the intervals between initial and midpoint, and between midpoint and final point.
However, as you show, if the graph is linear the result would be the same both ways.