The first part of this lesson involves a study of the relationship between the shape of a p-t graph and the motion of the object. If the position-time data for such a car were graphed, then the resulting graph would look like the graph at the right. Note that a motion described as a constant, positive velocity results in a line of constant and positive slope when plotted as a position-time graph.
Note that a motion described as a changing, positive velocity results in a line of changing and positive slope when plotted as a position-time graph. The position vs. The shapes of the position versus time graphs for these two basic types of motion - constant velocity motion and accelerated motion i. The principle is that the slope of the line on a position-time graph reveals useful information about the velocity of the object.
It is often said, "As the slope goes, so goes the velocity. If the velocity is constant, then the slope is constant i. If the velocity is changing, then the slope is changing i.
If the velocity is positive, then the slope is positive i. This very principle can be extended to any motion conceivable. Consider the graphs below as example applications of this principle concerning the slope of the line on a position versus time graph. The graph on the left is representative of an object that is moving with a positive velocity as denoted by the positive slope , a constant velocity as denoted by the constant slope and a small velocity as denoted by the small slope.
The graph on the right has similar features - there is a constant, positive velocity as denoted by the constant, positive slope. However, the slope of the graph on the right is larger than that on the left. This larger slope is indicative of a larger velocity. The object represented by the graph on the right is traveling faster than the object represented by the graph on the left. Students should be able to read the net displacement, but they can also use the graph to determine the total distance traveled.
Then ask how the speed or velocity is reflected in this graph. Direct students in seeing that the steepness of the line slope is a measure of the speed and that the direction of the slope is the direction of the motion.
Ask students where there zero should be. Why would the graph look different? What might account for the difference? Ask them to determine and compare average speeds for each interval.
What were the absolute differences in speeds, and what were the percent differences? Do the differences appear to be random, or are there systematic differences? Why might there be systematic differences between the two sets of measurements with different individuals in each role? So how do we use graphs to solve for things we want to know like velocity? Find the average velocity of the car whose position is graphed in Figure 1. The slope of a graph of d vs. Since the slope is constant here, any two points on the graph can be used to find the slope.
They can use whichever points on the line are most convenient. But what if the graph of the position is more complicated than a straight line? What if the object speeds up or turns around and goes backward?
Can we figure out anything about its velocity from a graph of that kind of motion? The graph in Figure 2. The graph of position versus time in Figure 2. The slope of the curve becomes steeper as time progresses, showing that the velocity is increasing over time. The slope at any point on a position-versus-time graph is the instantaneous velocity at that point.
It is found by drawing a straight line tangent to the curve at the point of interest and taking the slope of this straight line. Tangent lines are shown for two points in Figure 2. The average velocity is the net displacement divided by the time traveled. Calculate the instantaneous velocity of the jet car at a time of 25 s by finding the slope of the tangent line at point Q in Figure 2. The slope of a curve at a point is equal to the slope of a straight line tangent to the curve at that point.
The entire graph of v versus t can be obtained in this fashion. A curved line is a more complicated example. Define tangent as a line that touches a curve at only one point. Show that as a straight line changes its angle next to a curve, it actually hits the curve multiple times at the base, but only one line will never touch at all.
This line forms a right angle to the radius of curvature, but at this level, they can just kind of eyeball it. The slope of this line gives the instantaneous velocity. The most useful part of this line is that students can tell when the velocity is increasing, decreasing, positive, negative, and zero. Calculate the average velocity of the object shown in the graph below over the whole time interval.
Which of the following information about motion can be determined by looking at a position vs. If students are struggling with a specific objective, the Check Your Understanding will help identify direct students to the relevant content.
As an Amazon Associate we earn from qualifying purchases. Want to cite, share, or modify this book? Time Graphs. We Would Like to Suggest Sometimes it isn't enough to just read about it. You have to interact with it! And that's exactly what you do when you use one of The Physics Classroom's Interactives.
We would like to suggest that you combine the reading of this page with the use of our Graph That Motion or our Graphs and Ramps Interactives. Each is found in the Physics Interactives section of our website and allows a learner to apply concepts of kinematic graphs both position-time and velocity-time to describe the motion of objects.
Next Section: Determining the Slope on a p-t Graphs. Time Graphs » Meaning of Slope for p-t Graphs.
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