EXPLORATION 2.3B – Connecting veloc

EXPLORATION 2.3B – Connecting velocity and displacement using graphs
As we have investigated already with position-versus-time graphs, another way to represent motion is to use graphs, which can give us a great deal of information. Let’s now explore a velocity-versus-time graph, for the case of a car traveling at a constant velocity of +25 m/s.
Step 1 - How far does the car travel in 2.0 seconds? The car is traveling at a constant speed of 25 m/s, so it travels 25 m every second. In 2.0 seconds the car goes 25 m/s " 2.0 s, which is 50 m.
Step 2 – Sketch a velocity-versus-time graph for the motion. What on the velocity-versus-time graph tells us how far the car travels in 2.0 seconds? Because the velocity is constant, the velocity-versus-time graph is a horizontal line, as shown in Figure 2.11.
To answer the second question, let’s re-arrange Equation 2.2, , to solve for the displacement from the average velocity.
. (Equation 2.5: Finding displacement from
average velocity)
When the velocity is constant, the average velocity is the
value of the constant velocity. This method of finding the
displacement can be visualized from the velocity-versus-time graph.
The displacement in a particular time interval is the area under the velocity-versus-time graph for that time interval. “The area under a graph” means the area of the region between the line or curve on the graph and the x-axis. As shown in Figure 2.12, this area is particularly easy to find in a constant-velocity situation because the region we need to find the area of is rectangular, so we can simply multiply the height of the rectangle (the velocity) by the width
of the rectangle (the time interval) to find the area (the displacement).
Key idea: The displacement is the area under the velocity-versus-time graph. This is true in general, not just for constant-velocity motion.
Substitute Equation 2.1, .
This gives:
, into Equation 2.5,
.
Figure 2.12: The area under the velocity- versus-time graph in a particular time interval equals the displacement in that time interval.
Figure 2.11: The velocity-versus- time graph for a car traveling at a constant velocity of +25 m/s.
Deriving an equation for position when the velocity is constant
Generally, we define the initial time to be zero: .
Remove the “f” subscripts to make the equation as general as possible: .
(Equation 2.6: Position for constant-velocity motion) Related End-of-Chapter Exercises: 3, 17, and 48.
Essential Question 2.3: What are some examples of real-life objects experiencing constant- velocity motion? (The answer is at the top of the next page.)
.
Such a position-as-a-function-of-time equation is known as an equation of motion.
Chapter 2 – Motion in One Dimension Page 2 - 7