Scope, Sequence, and Coordination
A Framework for High School Science Education
Based on the National Science Education Standards
Air Resistance and Friction
Before students can understand how a net force is connected to motion, they need to understand how motion is described. This topic is called kinematics. Motion in one direction along a straight line involves just the ordinary concept of distance, as might be read on an odometer. For such a motion we can use scalar distances, speeds, and accelerations.
If, however, the motion reverses or changes direction, the concept of distance becomes more complicated. The actual distance along the path would be more than the final straight-line distance from the starting point. In such cases, we need to introduce the concept of displacement. For motion along a curved path, we must introduce the concept of a vector displacement. However, for motion along a straight line, the plus or minus sign can be used to indicate direction. This approach allows for much of kinematics to be studied before introducing two- or three-dimensional motion, where the vector character of motion becomes essential.
When first studying motion, students must learn the subtle distinctions between time (as a clock reading) and a time interval. They must understand that average speed is a definition of total distance traveled (as might be indicated on an odometer) divided by how long the motion occurs, and not always some average of initial and final values of speed, the way ordinary averages are often computed.
Students need to construct and interpret graphs to learn that the slope of a d-t curve gives speed and that the slope of a v-t curve gives acceleration. They also need to learn that the "area" under a v-t curve gives distance traveled. Using these graphical techniques, they can solve complex motion problems without the use of equations or formulas.
At the more advanced levels, where there is two- and three-dimensional motion, students learn that we must use displacement, velocity, and vector acceleration (instead of simply distance, speed, and scalar acceleration). They will learn that these quantities do not add, subtract, or multiply using ordinary arithmetic or algebraic operations, and do not even use all of the rules of the arithmetic or algebra of real numbers (vector products are not commutative, for example). Instead, they must use vector arithmetic and algebra. This mathematics motion along curved paths is described quantitatively, like motion along circular or parabolic paths, and it leads to concepts like centripetal acceleration.
Students must also learn that complex motions can be considered by examining components along the axes of a coordinate system, recognizing that motion along the x, y, and z axes can be considered independently of one another. For example, motion in two dimensions can be considered as two one-dimensional problems using scalars; then the motion can be considered again as a vector using these two components. This is especially important in describing projectile motion and in understanding the concepts of a reference frame and relative velocity.
In working with slopes of a d-t or v-t graph to arrive at speeds or accelerations when speed in the case of a d-t curve, or acceleration in the case of the v-t curve, is not constant, we must take very small intervals of time to approximate a tangent to the curve. This, of course, was the problem that led Newton to invent differential calculus. A derivative finds the slope in the limit as the time interval approaches zero in its duration.
Instant in time (clock reading), interval of time, reference point, position, distance, change in position, average speed, instantaneous speed (as a speedometer reading only), average acceleration
Air resistance, reference frame, projectile motion, circular motion, centripetal acceleration
Vector displacement, velocity, instantaneous velocity, change in velocity, acceleration and instantaneous acceleration
Combining vector quantities, vector components,
AArea@ under a speed vs. time (v-t) curve gives distance, if motion is in one direction along a path. If the direction reverses during the motion, the Aarea@ under the velocity vs. time curve gives the displacement. Similarly, for motion in one direction along a path, the slope of a distance vs. time (d-t) curve gives speed, and the slope of a speed vs. time curve gives the scalar acceleration along the path. But if the motion involves a direction reversal, the slope of a displacement time graph gives the velocity, and the slope of a velocity time graph gives the acceleration (as a vector).
Model of an extended object moving, but represented by a particle (point) in motion