MW5 Analyze graphically and mathematically the relationship among distance, speed, time, and acceleration for objects that undergo simple linear motion or uniformly accelerated motion
Suggested time: 4-6 hours
The focus of this objective is for students to develop skills in the graphical and quantitative analysis of motion. The depth with which a teacher explores graphical and quantitative analysis will depend considerably on the mathematical abilities of the students, but it should be sufficient to limit the use of motion equations to those listed below. Students should engage in limited problem solving using the equations of motion.
It is not necessary to develop the equations of linear motion directly from student graphs, but doing so will likely increase student understanding that graphs and equations are two different but related methods of representing the motion of an object. A common misconception among students is that the shape of a graph is a physical representation of the actual motion (e.g., a ball rolling down a ramp would result in a graph that slopes downward).
Students should analyze graphs using a range of appropriate technologies including graph paper and ruler, graphing calculators, and graphical analysis software. Teachers may choose to integrate this objective with the previous two foundational objectives.
Learning Objectives
- Describe quantitatively the relationship among distance, time, speed, and acceleration for everyday objects that undergo simple linear motion (uniform motion or uniformly accelerated motion).
- Identify the physical quantity that the slope of a distance-time graph represents.
- Identify the physical quantity that the slope of a speed-time graph and the area under a speed-time graph represent.
- Solve problems related to uniform motion and uniformly accelerated motion using the equations of motion.
- Use distance-time and speed-time graphs to solve problems related to uniform motion and uniformly accelerated motion. (NUM)
- State a prediction and a hypothesis based on available evidence and background information when solving problems relating to simple linear motion.
- Compare theoretical and empirical values and account for discrepancies. (CCT)
- Read and interpret graphs to develop an understanding of the relationships among numbers. (NUM)
Enrichment Learning Objectives
- Describe quantitatively the relationship among displacement, time, velocity, and acceleration for everyday objects that undergo simple linear motion (uniform motion or uniformly accelerated motion).
- Use position-time and velocity-time graphs to solve problems related to uniform motion and uniformly accelerated motion.
Key Questions
- How do distance-time and speed-time graphs represent the motion of objects?
- How do uniform motion and uniformly accelerated motion appear on distance-time graphs?
- How do uniform motion and uniformly accelerated motion appear on speed-time graphs?
- How are graphical and mathematical methods of representing an object's motion related?
Key Concepts
- Uniform motion appears as a straight, sloped line on a distance-time graph and as a horizontal line on a speed-time graph.
- The slope of a distance-time graph represents the speed of the object.
- A useful equation for uniform motion is v = Δd/Δt.
- Uniformly accelerated motion appears as a parabola on a distance-time graph and as a straight, sloped line on a speed-time graph.
- The slope of a speed-time graph represents the acceleration of the object.
- The area under a speed-time graph represents the change in distance of the object.
- Some useful equations for uniformly accelerated motion are:
a = Δv/Δt
Δd = v i t + 1/2aΔt2
- Numbers can convey important information in science and may be used to express physical relationships in an abstract format.
Pre-Instructional Questions
- Do students understand the relationship between graphs and equations for representing uniform motion or uniformly accelerated motion?
- Which motion equations have students previously used?
Suggested Teaching Strategies and Activities
- Students should analyze graphs of uniform motion created during previous experiments and identify the characteristics of distance-time and speed-time graphs (i.e., the distance-time graph is a straight, sloped line; the speed-time graph is a horizontal line).
- Students should analyze graphs of uniformly accelerated motion created during previous experiments and identify the characteristics of distance-time and speed-time graphs (i.e., the distance-time graph is a parabola; the speed-time graph is a straight, sloped line).
- Students should draw generalized distance-time and speed-time graphs to represent objects exhibiting no motion, uniform positive (forward) motion, uniform negative (backward) motion, uniformly accelerated motion, and uniformly decelerated motion. These could be separate graphs or sections within the same graph.
- Students should construct a distance-time and velocity-time graph of an object that exhibits uniform motion. They should then calculate the slope of the distance-time graph and compare that value with the value of the velocity on the speed-time graph for the same time interval to see that these values are numerically equal, within margins of uncertainty.
- Students should explain the similarities and differences between various representations of motion (written, visual, graphical, and numerical) and the advantages and disadvantages of each method.
- Students should be able to analyze motion graphs (distance-time and speed-time) and explain why the data points do or do not lie entirely along the best-fit curve or line. As part of this discussion, students should identify what a best-fit line is and how that line relates to the equations of motion and the actual motion of the object. Students should be able to explain reasons why actual values of a variable such as distance or speed are not always identical to values predicted from one or more of the equations of motion.
- Students should use distance-time graphs of objects undergoing uniform motion to:
- Explain how to calculate the average speed of the object graphically.
- Calculate the average speed between any two points on the graph.
- Explain how to calculate the instantaneous speed of the object graphically.
- Calculate the instantaneous speed at any given point on the graph.
- Explain the similarities and differences between average and instantaneous speed calculations.
- Determine the distance traveled by the object at any given instant.
- Graphically determine the distance traveled by the object during any given time interval.
- Graphically determine the time elapsed while the object moved through various distances.
- Students should use speed-time graphs of objects undergoing uniformly accelerated motion to:
- Explain how to calculate the average acceleration of the object graphically.
- Calculate the average acceleration between any two points on the graph.
- Explain how to calculate the instantaneous acceleration of the object graphically.
- Calculate the instantaneous acceleration at any given point on the graph.
- Determine the speed of the object at any given instant.
- Explain how to calculate the distance traveled by the object during any given time interval.
- Graphically determine the distance traveled by the object during any given time interval.
Throughout these activities, students are required to locate, interpret, evaluate, and use life/work information. (CD 5.3)