Describing Motion with Position vs. Time Graphs

The specific features of the motion of objects are demonstrated by the shape and the slope of the lines on a position vs. time graph.

To begin, consider a car moving with a constant, rightward (+) velocity of 10 m/s.

Note that a motion with constant, positive velocity results in a line of constant and positive slope when plotted as a position-time graph.

The Principle of Slope for a position–time Graph

The shapes of the position vs. time graphs for these two basic types of motion – constant velocity motion and changing velocity motion (i.e.accelerated motion) – reveal an important principle.

The principle is that the slope of the line on a position-time graph reveals useful information about the velocity of the object. It's often said, "As the slope goes, so goes the velocity."

If the velocity is constant, then the slope is constant (i.e., a straight line). If the velocity is changing, then the slope is changing (i.e., a curved line). If the velocity is positive, then the slope is positive (i.e., moving upwards and to the right). If the velocity is negative, then the slope is negative (i.e., Moving downward to the right).

Slow, Rightward (+) Constant Velocity

Positive Velocity Zero Acceleration

Fast, Rightward (+) Constant Velocity

Slow, Leftward (–) Constant Velocity

Fast, Leftward (–) Constant Velocity

Leftward (–) Velocity; Slow to Fast

Leftward (–) Velocity; Fast to Slow

Now consider a car moving with a changing, rightward (+) velocity – that is, a car that is moving rightward and speeding up or accelerating.

Note that a motion with changing, positive velocity results in a line of changing and positive slope when plotted as a position-time graph.

Positive Velocity Constant Velocity

Positive Velocity Changing Velocity (acceleration)

The Passing Lane position vs. time graph

The Passing Lane Velocity vs. Time Graph

The Stoplight

Two-Stage Rocket

Consider a car moving with a constant velocity of +10 m/s for 5 seconds. The diagram below depicts such a motion.

Consider a car moving with a constant velocity of +5 m/s for 5 seconds, stopping abruptly, and then remaining at rest (v = 0 m/s) for 5 seconds.

1. Consider the graph at the right. The object whose motion is represented by this graph is ... (include all that are true):

a. moving in the positive direction.b. moving with a constant velocity.c. moving with a negative velocity.d. slowing down. e. changing directions.f. speeding up.

g. moving with a positive acceleration.h. moving with a constant acceleration.

Determining the Area on a velocity-time Graph

1. A plot of velocity vs. time can be used to determine the acceleration of an object (slope = acceleration).

2. A plot of velocity vs. time can also be used to determine the distance traveled by an object

3. For velocity vs. time graphs, the area bounded by the line and the axes represents the distance traveled.

The shaded area is representative of the distance traveled by the object during the time interval from 0 seconds to 6 seconds. This representation of the distance traveled takes on the shape of a rectangle whose area can be calculated using the appropriate equation.

The shaded area is representative of the distance traveled by the object during the time interval from 0 seconds to 4 seconds. This representation of the distance traveled takes on the shape of a triangle whose area can be calculated using the appropriate equation.

The shaded area is representative of the distance traveled by the object during the time interval from 2 seconds to 5 seconds. This representation of the distance traveled takes on the shape of a trapezoid whose area can be calculated using the appropriate equation.

Calculating the Area