Lesson 4: Describing Motion
with Velocity vs. Time Graphs
Determining the Area on a v-t
Graph
As you learned in an earlier section of
this lesson, a plot of velocity vs. time can be used to
determine the acceleration of an object (slope = acceleration).
In this part of the lesson, you will learn how a plot of
velocity vs. time can also be used to determine the
distance traveled by an object. For velocity vs. time
graphs, the area bounded by the line and the axes
represents the distance traveled.
The diagram below shows three
different velocity-time graphs; the shaded regions
between the line and the axes represent the
distance traveled during the stated time interval.
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.
The method used to find the area under
a line on a velocity-time graph depends on whether the
section bounded by the line and the axes is a rectangle, a
triangle or a trapezoid. Area formulae for each shape are
given below.
Calculating
the Area of a Rectangle
The shaded rectangle on the
velocity-time graph, below, has a base of 6 s and a height of 30
m/s.
Area of rectangle: A = b x h = (6 s) x (30 m/s) = 180 m.
The object was displaced 180 meters during the first
6 seconds of motion.
Area = b *
h
Area = (6 s) * (30 m/s)
Area = 180 m
Now check your understanding by finding the distance
traveled by the object in each of the following cases.
Calculating
the Area of a Triangle
The shaded triangle on the velocity-time graph, below, has a base of 4
seconds and a height of 40 m/s.
Area of triangle: A = 0.5 * b * h = (0.5) * (4 s) * (40 m/s) = 80 m.
The object was displaced 80 meters during the first four seconds of
motion.
Area = 0.5 * b *
h
Area = (0.5) * (4 s) *
(40 m/s)
Area = 80 m
Check your understanding by finding the distance traveled
by the object in each of the following cases.
Calculating
the Area of a Trapezoid
The shaded trapezoid on the velocity-time graph, below, has a base of 2
seconds and heights of 10 m/s (on the left side) and 30
m/s (on the right side).
Area of trapezoid: A = (0.5) * (b) * (h1 + h2) = (0.5) * (2 s) * (10 m/s + 30
m/s) = 40 m.
The object was displaced 40 meters during the time interval
from 1 second to 3 seconds.
Area = 0.5 * b *
(h1 + h2)
Area = (0.5) * (2 s) * (10 m/s + 30 m/s)
Area = 40 m
Now check your understanding by finding the distance traveled
by the object in each of the following cases.
Alternative
Method for Calculating the Area of a Trapezoid
An alternative method of determining the area of a
trapezoid involves breaking the trapezoid into a triangle
and a rectangle. The areas of the triangle and rectangle
are computed individually; the area of the trapezoid
is then the sum of the areas of the triangle and the
rectangle. This method is illustrated below.
Triangle: Area = (0.5) * (2 s) *
( 20 m/s) = 20 m
Rectangle: Area = (2 s) * (10
m/s) = 20 m
Trapezoid: Area = 20 m + 20 m = 40 m
To review, in this
lesson you learned that the area bounded by the line and the axes of
a velocity-time graph is equal to the displacement of an
the object during that time interval. The shaded region can
be identified as either a rectangle, triangle, or
trapezoid whose area can subsequently be determined using
the appropriate formula. Once calculated, this area
represents the displacement of the object during the time period indicated.