Consider an object undergoing motion along a straight-line path, where the
motion is characterized by a few consecutive time intervals during each of which
the acceleration is constant but typically at a different constant value than it is for
the adjacent specified time intervals. The acceleration undergoes abrupt changes
in value at the end of each specified time interval. The abrupt change leads to a
jump discontinuity in the Acceleration vs. Time Graph and a discontinuity in the
slope (but not in the value) of the Velocity vs. Time Graph (thus, there is a
“corner” or a “kink” in the trace of the Velocity vs. Time graph). The thing is,
the trace of the Position vs. Time graph extends smoothly through those instants
of time at which the acceleration changes. Even folks that get quite proficient at
generating the graphs have a tendency to erroneously include a kink in the
Position vs. Time graph at a point on the graph corresponding to an instant when
the acceleration undergoes an abrupt change.
Your goals here all pertain to the motion of an object that moves along a straight line path at a
constant acceleration during each of several time intervals but with an abrupt change in the value
of the acceleration at the end of each time interval (except for the last one) to the new value of
acceleration that pertains to the next time interval. Your goals for such motion are:
(1) Given a description (in words) of the motion of the object; produce a graph of position vs.
time, a graph of velocity vs. time, and a graph of acceleration vs. time, for that motion.
(2) Given a graph of velocity vs. time, and the initial position of the object; produce a description
of the motion, produce a graph of position vs. time, and produce a graph of acceleration vs.
time.
(3) Given a graph of acceleration vs. time, the initial position of the object, and the initial
velocity of the object; produce a description of the motion, produce a graph of position vs.
time, and produce a graph of velocity vs. time.
The following example is provided to more clearly communicate what is expected of you and
what you have to do to meet those expectations:
Example 9-1
A car moves along a straight stretch of road upon which a start line has been
painted. At the start of observations, the car is already 225 m ahead of the start line
and is moving forward at a steady 15 m/s. The car continues to move forward at
15 m/s for 5.0 seconds. Then it begins to speed up. It speeds up steadily, obtaining
a speed of 35 m/s after another 5.0 seconds. As soon as its speed gets up to
35 m/s, the car begins to slow down. It slows steadily, coming to rest after another
10.0 seconds. Sketch the graphs of position vs. time, velocity vs. time, and
acceleration vs. time pertaining to the motion of the car during the period of time
addressed in the description of the motion. Label the key values on your graphs of
velocity vs. time and acceleration vs. time.
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