(Have ready the supplies [toy cars, ball, incline, dynamics cart] to present the four motion scenarios, plus motion detectors with their necessary software and/or interfaces, as described in more detail in the Lesson Background section.)
In this lesson, you will observe moving objects and discuss position, velocity and acceleration to describe motion. The four different scenarios of moving objects are:
Two toy cars that move across a table or floor with constant speeds, one faster than the other.
A ball that speeds up at a uniform rate as it rolls down an incline.
A dynamics cart that slows down at a uniform rate as it rolls across a table or floor.
A person walking across the room with a speed that changes irregularly.
For each scenario, observe the moving objects and sketch predicted position vs. time and velocity vs. time graphs for each. After you observe all the examples, consider these questions. (Refer to Table 1; read the questions aloud, write them on the classroom board, or show the class the
Six Questions Visual Aid
(Proceed to demonstrate the four scenarios in the classroom, directing students to sketch predicted graphs for each and then answer the questions in Table 1.)
Now, using a motion detector, interface and software, observe each moving object
, while collecting data to generate position vs. time and velocity vs. time graphs as the objects are moving. To accomplish this, use a sonar-based motion detector. These devices measure where an object is located as long as it is directly in front of the sensor and nothing between the object and the sensor blocks the sound waves. Then use software to interpret the data collected using the motion detector. Finally, compare your predicted graphs to the graphs produced using the motion detector's data and discuss any differences.
Lesson Background and Concepts for Teachers
To collect data for generating position vs. time and velocity vs. time graphs, have students use sonar-based Vernier motion detectors or similar devices. These sensors require software to interpret the data. With the Vernier device, use Logger Pro, or Logger Lite—a free download. Some motion detectors also require an interface, but Vernier has a version that connects directly to a computer via USB. Vernier also has a CBR version that connects directly to a compatible TI-calculator and uses internal software to record data.
Possible motion detector options:
Possible interface options:
Possible software options:
As students compare their predicted graphs to the graphs produced using the motion detector data, the ultimate goal is for them to understand that the slope of a tangent line at a given point is the object's instantaneous velocity and that a velocity vs. time graph is just a representation of an object's instantaneous velocities over time. If necessary, guide the class discussion so that students reach this understanding.
In order to complete the associated activity,
"Gaitway" to Acceleration: Walking Your Way to Acceleration
, students must understand what a secant line to a curve is and how to compute Riemann sums. So, teach students the following lesson content to prepare them for the associated activity.
: A secant line of a curve is a line that intersects a curve in a local region at two points on the curve. As the two intersection points become closer together on the curve, the secant line becomes closer and closer to the tangent line at a point on the curve. In calculus, the derivative evaluated at a point on the curve is the slope of the tangent line at that evaluated point. A secant line is a way to approximate derivatives without taking a derivative.
In the associated activity, the data does not have a corresponding equation (although you could perform a regression to find one) so taking a derivative is not possible. Secant lines allow the approximation of the derivative (which would represent the velocity of the object) without requiring the computation of the derivative. If you create a curve from the associated points found by taking a derivative (or approximating using secant lines), you can create a velocity curve of the object. Computing secant lines for this curve in the same fashion as the previous example is a method for approximating the second derivative, which represents the acceleration of the object. Again, by using secant lines, the acceleration can be approximated without having an equation and using calculus. To compute a secant line, select two points, calculate the slope, plug one of the selected points and the slope into point slope form, and then algebraically manipulate it into any form of the line that you wish. When working from the object's position, the secant line evaluated at an appropriate "x" value yields a "y" value that represents the object's velocity (first derivative). When working from the object's velocity, the secant line evaluated at an appropriate "x" value yields a "y" value that represents the object's acceleration (second derivative).
: A Riemann sum is an approximation of the area under a curve. The sum is computed by dividing the region into polygons (rectangles, trapezoids, etc.) that when combined approximate the area under the curve. The area for each of the polygons is computed using an appropriate area equation and the results are added to approximate the region. Using Riemann sums, a numerical approximation of a definite integral can be found. Similar to the secant line, a Riemann sum can be used to approximate an object's velocity or position without having an equation that you can integrate. An integral is the inverse of a derivative. Hence, a Riemann sum approximation works backwards from a secant line approximation. Given an object's acceleration curve, a Riemann sum can be used to determine an object's velocity curve. Given an object's velocity curve for an object, a Riemann sum can be used to determine an object's position curve.
Various Definitions of Acceleration
In recognizable terms
: In common words, acceleration is a measure of the change in speed of an object, either increasing (acceleration) or decreasing (deceleration). This definition is not completely accurate because it disregards the directional component of the velocity vector. Vectors have two components—magnitude and direction. When discussing speed, we only consider the change in magnitude.
In conceptual terms:
Acceleration is a quantity in physics that is defined to be the rate of change in the velocity of an object over time. Since velocity is a vector, acceleration describes the rate of change in the magnitude and direction of the velocity of an object. When thinking in only one dimension, acceleration is the rate that something speeds up or slows down.
In mathematical terms:
Many different mathematical variations exist for acceleration. Below is a partial listing:
Newton's second law of motion
: For a body with constant mass, the acceleration is proportional to the net force acting on it. F
Rate of change in velocity
with respect to time, slope of velocity vs. time graph (two forms):
is when the velocity of an object in motion changes by an equal amount in equal interval time periods. Using algebra, the following kinematic equations can be derived:
In process terms
: To compute the acceleration of an object, it is first essential to understand what type of motion is occurring. Once the type of motion is determined, a variety of mathematical equations can be applied, depending on the situation. Unfortunately, the acceleration is only easy to find in situations in which the object's motion is predictable. For instance, when an object is undergoing harmonic motion, the acceleration of the object can be determined because the object's position is predictable at any point in time.
In applicable terms
: Any object in motion has acceleration. If the object's velocity is changing, the object is either accelerating or decelerating. If the object has constant velocity, the object's acceleration is zero. If an object is moving at a constant speed following a circular path, the object experiences a constant acceleration that points toward the center of the circle.