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Chapter 2: Kinematics: Description of Motion
Outline
2.1 Distance and Speed: Scalar Quantities
2.2 One-Dimensional Displacement and Velocity: Vector Quantities
2.3 Acceleration
2.4 Kinematic Equations (Constant Acceleration)
2.5 Free Fall
Summary
Chapter 2 begins the study of motion. This chapter concentrates on one-dimensional kinematics, or the study of motion in a straight line without regard to its causes. Vectors and scalars are introduced, but the vectors are limited to one dimension so that they can be assigned positive and negative values to indicate direction. Chapter 2 lays the foundation for twodimensional motion, which is addressed in Chapter 3.
Major Concepts
By the end of the chapter, students should understand each of the following and be able to demonstrate their understanding in problem applications as well as in conceptual situations.
· Scalar quantities and vector quantities
· Position, distance, and displacement
· Speed and velocity
§ Average
§ Instantaneous
§ Constant
· Acceleration
§ Average
§ Instantaneous
§ Constant
· Graphs of position versus time, velocity versus time, and acceleration versus time
· Kinematic equations (constant acceleration)
· Free fall
§ Acceleration due to gravity, g
§ Kinematic equations for free fall
Teaching Suggestions and Demonstrations
Many of the concepts in this chapter, such as velocity and acceleration, are familiar to students from everyday experiences like driving a car. Most students, however, will not know the physics definitions of these same terms or how they relate to one another. In addition, common misconceptions due to the presence of air resistance and friction in the real world need to be addressed. Because problem solving is a large component of most physics courses, it is a good idea to emphasize it from the beginning. A short quiz at the end of this chapter can be helpful at this early stage to give students practice solving problems on their own.
Sections 2.1 and 2.2
Beginning physics students already know what distance and speed are; however, you will need to help them differentiate between distance and displacement and between speed and velocity. Introduce vectors and scalars with some examples. For instance, if a student travels 100 km due east, stops for lunch, and then travels 20 km farther due east, she will be 120 km from home. If a second student travels 100 km due east, stops and picks up a friend, and then turns around and drives 20 km due west, he will be only 80 km from his starting point. A third student may travel 100 km due east, then make a left turn and travel 20 km due north. All three have different displacements although each student has traveled the same distance. This should illustrate to students that vectors don’t add as numbers do. Introduce vectors in the general sense, including two-dimensional examples, but then point out that Chapter 2 deals with one-dimensional motion only.
It is important to distinguish among constant, average, and instantaneous speed and velocity. Example 2.2 and the associated follow-up exercise emphasize the difference between average speed and average velocity. Figure 2.8 helps illustrate the difference between instantaneous and average velocities.
Use Ranking Task Exercise “Ball Motion Diagrams—Velocity I1” to check for student understanding of velocity before moving on to acceleration. As students begin working with acceleration they will need frequent reminders that the familiar “distance equals speed times time” is really “distance equals average speed times time.”
Sections 2.3 and 2.4
The units of acceleration are initially puzzling. Have students calculate a common acceleration, such as the acceleration of a car on the entrance ramp to a highway. They may initially use units that are intuitively sensible, such as miles per hour per second, and then convert to the more conventional “meters per second squared” to get a feel for the units they will be using in physics class.
Students often have trouble understanding the important differences between constant velocity and constant acceleration the first time they are introduced to these concepts. A helpful approach is to directly compare these two cases. Imagine, for instance, a cart with a spark timer that leaves marks on a straight track at one-second intervals. Students can determine distance traveled by the cart during each second when it is moving with a constant velocity and then again when it is moving with a constant acceleration. When the distances are plotted on a one-dimensional line, it is apparent that the cart covers equal distances in each one-second interval for the constant velocity case but increasingly larger distances in each successive second for the constant acceleration case. Do Ranking Task Exercise “Ball Motion Diagrams—Acceleration I2” with the students at this point and compare to the previous exercise. The diagrams in the two exercises are the same, but one asks for information about velocity and the other about acceleration. You can follow up with Ranking Task Exercises “Ball Motion Diagrams—Velocity II3” and “Ball Motion Diagrams—Acceleration II4.” These exercises provide students with practice in using different coordinate systems.
Graphs can often be difficult for beginning physics students to interpret. The preceding exercise, in which the position of a car each second is plotted on a straight line, is a nice intermediate step leading into actual position-versus-time graphs, as illustrated by Figure 2.6. Ranking Task Exercise “Position Time Graphs—Displacement8” will help students with graph interpretation as well as with the difference between distance and displacement. You can also create the corresponding velocity- and acceleration-versus-time graphs and determine the significance of slopes and areas under the curves. Spend some time with Figures 2.7, 2.8, and 2.10 to help students with graphical interpretation. Relating the three graphs of position, velocity, and acceleration versus time to one another and to the description of the motion requires practice and is a very valuable exercise whether used quantitatively or conceptually.
Demo 2-1 A motion sensor with computer interface and software for plotting the graphs of actual moving objects is an excellent demonstration tool for graphical understanding and interpretation. (See Resource Information.) One approach is to have students sketch their predictions for a certain motion and then receive immediate feedback from the computer. (For instance, ask them to sketch the three graphs for a ball thrown straight up into the air and then use the motion sensor to plot the graphs.) Alternatively,
students can try to duplicate a graph shown on the computer by moving an object or themselves in front of the motion sensor. Duplicating the shape of a velocity-versus-time graph is particularly challenging!
Throughout this chapter, the significance of positive and negative signs needs to be emphasized. The meanings of positive and negative displacement and velocity are directly related to the vector nature of these quantities. The same is true of acceleration, but the direction, and therefore the sign, of acceleration is more confusing. Use the Learn By Drawing in Section 2.3 and do some numerical examples to help convince students that a negative acceleration does not necessarily mean an object is slowing down. (An object slowing down while traveling in the positive direction and an object speeding up while traveling in the negative direction both have negative accelerations.)
Lots of practice is required for most students to achieve a solid understanding of the kinematic equations for constant acceleration, Equations 2.8 through 2.12. Initially, students may need guidance in using the equations and in recognizing the various variables. Point out which variables are scalars and which are vectors. Also discuss which quantities stay constant for a given problem and which depend on time. Work many examples, and encourage students to try problems on their own as well. Conceptual Example 2.6 examines the relationship between final velocity and time for a constant acceleration. Figure 2.13 provides a connection between the kinematic equations and the relationship between velocity and acceleration expressed in graphical form.
Section 2.5
Free fall provides a wonderful opportunity for emphasizing major points regarding position, velocity, and acceleration for the case of constant acceleration. Begin by discussing g, the acceleration due to gravity, and defining free fall. It is worth spending some time on the fact that in the absence of air resistance, g is a constant. Insight 2.1 on Galileo gives some interesting background about the famous scientist and the Leaning Tower of Pisa.
Demo 2-2 The simple demonstration of dropping objects simultaneously from the same height helps convince students that acceleration due to gravity really is the same for all objects. Choosing a wide variety of objects can also help them get a feel for the conditions under which air resistance can and cannot be ignored.
Demo 2-3 One very simple but effective demonstration is to drop a quarter or a ball and a piece of notebook paper simultaneously from the same height. First, give the class a chance to guess what they think will happen. They have just learned that the acceleration due to gravity is the same for all objects, but they will usually revert to their experience and guess that the quarter will fall first. Then, drop the quarter and the paper. Discuss with the class why the quarter fell first. Someone will mention air resistance. Discuss ways in which air resistance could be taken out of the problem. Crush the paper into a ball and again poll the class about which object will hit the ground first. This time about half the class will guess that the two objects will fall at the same rate. Drop the quarter and the paper and discuss the results. Students respond well to this demonstration; it gives them dramatic evidence of the role of air resistance in free fall and information about when it can be ignored.
Demo 2-4 Another very simple demonstration involves filling a Styrofoam (or paper) cup with water and punching a hole with a pencil in the side near the bottom. (It is best to do this over a sink or a trash can, since it can be messy.) When the cup is held stationary, water shoots out of the hole. (A nice discussion about the shape of the curve the water makes can be led if the cup is large enough to hold sufficient water.) When the cup is dropped, water will not come out of the hole, since the cup and the water are falling at the same rate. Be sure to ask the class what they think will happen before dropping the cup; often, no one will guess that no water will emerge from the hole.
Once students are convinced of the constancy of g, turn the kinematic equations for constant acceleration into free-fall equations by replacing a with –g and changing the x’s to y’s. Introductory physics students are often overwhelmed by the number of equations thrown at them at the beginning of the term, so try to convince them that these equations aren’t new ones; they are just revised for a special case of constant acceleration due to gravity. In Ranking Task Exercise “Vertical Model Rockets—Maximum Height6,” both mass and initial velocity of each projectile are given, and students need to recognize that the mass is irrelevant.
Although most students understand intuitively that the velocity of a ball thrown up in the air is zero at the top of its trajectory, many assume incorrectly that the acceleration there is zero as well. Graph y versus t, v versus t, and a versus t for a problem such as Example 2.11 in order to correct misconceptions about acceleration for objects in free fall. Remind students that acceleration is a rate of change. Even though the object moves first up and then down, the rate of change of its velocity is a constant –9.80 m/s².
Emphasize the importance of consistency with sign conventions, as can be demonstrated by comparing a problem in which an object is thrown straight up from a bridge to one in which an object is thrown straight down from the same bridge at the same speed. Try Example 2.9 and then expand on it to find the time it would take a stone thrown upward at 14.7 m/s to hit the water. Although the speeds are the same, the velocities have different signs and so are not the same, resulting in very different outcomes. Explicitly pointing out the positive direction at the outset of each problem reminds students of this important distinction.
Demo 2-5 A free-fall apparatus with spark timer can demonstrate change in position for objects undergoing acceleration due to gravity. The sparks mark the paper at equal time intervals, and therefore the distance between adjacent sparks increases as the plummet accelerates. This demonstration is useful for helping students visualize constant acceleration and can also be used to calculate the acceleration due to gravity. (See Resource Information.)
Demo 2-6 Students get a kick out of measuring their own reaction time and comparing it to that of others. Using free fall to measure reaction time is described in Example 2.10 and illustrated in Figure 2.15. You can have a couple of students demonstrate, or break the class into pairs and have everyone try it. As an added bonus, plot the reaction times from the whole class and use the graph to introduce bell curves and standard deviations.
Be sure to assign homework from the Pulling It Together section at the end of the chapter. We recommend problems 78, 79, and 82 in Chapter 2.
Resource Information
Suggested Ranking Task Exercises
Ball Motion Diagrams—Velocity I1
Ball Motion Diagrams—Acceleration I2
Ball Motion Diagrams—Velocity II3
Ball Motion Diagrams—Acceleration II4
Vertical Model Rockets—Maximum Height6