The diagram shows a graph of the
variation of the position 𝑥 of an object with time 𝑡. What is the velocity of the object
in the time interval 𝑡 greater than zero seconds to 𝑡 equal 0.5 seconds? What is the velocity of the object
in the time interval 𝑡 greater than 0.5 seconds to 𝑡 equals 1.0 seconds? What is the velocity of the object
in the time interval 𝑡 greater than 1.0 seconds to 𝑡 equals 2.0 seconds?
In this exercise, we want to solve
for three average velocities. We’ll call them 𝑣 one, 𝑣 two, and
𝑣 three. These average velocities are based
on our position versus time graph, where 𝑣 one applies to the first leg of the
journey, 𝑣 two applies to the second leg, and 𝑣 three applies to the third and
final leg of this movement. To get started, we can recall that
average velocity 𝑣 sub avg is equal to displacement divided by time 𝑡. Let’s apply this relationship to
solve first for 𝑣 sub one.
𝑣 sub one is equal to the position
of our object 𝑝 when time equals 0.5 seconds minus our object’s position when time
equals zero seconds. And this difference in position,
called the displacement, is then divided by the time it takes to move that
distance. Looking at our diagram, we can mark
out the position of the object at 𝑡 equals 0.5 seconds and 𝑡 equals zero
seconds. And if we drop a vertical line down
from our second point, we see that the time interval Δ𝑡 for all this to happen is
equal to 0.5 seconds. This means that 𝑣 sub one is 1.0
meters per second. That’s the average velocity of the
object over this initial time interval.
Next, we move on to calculating 𝑣
sub two. 𝑣 sub two is the average velocity
of the object between time equals 1.0 seconds and 0.5 seconds. Looking at the diagram of the
object’s position versus time, we see that its position at both these time values is
the same. It’s 0.5 meters. That means 𝑣 sub two is 0.0 meters
per second. That’s the average velocity of our
object over a time interval where its position doesn’t change.
Finally, we move on to calculating
𝑣 three, the average velocity of our object over the third time interval from time
𝑡 equals 2.0 seconds to 1.0 seconds. Looking on our diagram, we see that
at 2.0 seconds our object has a position of 0.0 meters and at 1.0 seconds it has a
position of 0.5 meters. Calculating this fraction, we find
𝑣 sub three is negative 0.50 meters per second. That’s the object’s average
velocity on the last leg of its movement.