Activity 5—Rolling Ball

Concepts

Function explored: parabolic

Plotting a ball rolling down a ramp of varying inclines creates a family of curves, which can be modeled by a series of quadratic equations. This activity investigates the values of the coefficients in the quadratic equation, y = ax2 + bx + c.

Materials

Ÿcalculator (see page 2 for available models)

ŸCBR 2™ motion detector

Ÿunit-to-CBR 2™ or I/O unit-to-unit cable

ŸEasyData application or RANGER program

Ÿlarge (9 inch) playground ball

Ÿlong ramp (at least 2 meters or 6 feet—a lightweight board works well)

Ÿprotractor to measure angles

Ÿbooks to prop up ramp

ŸTI ViewScreené panel (optional)

Hints

Discuss how to measure the angle of the ramp. Let students get creative here in measuring the initial angle. For example, they might use a trigonometric calculation or folded paper.

For steeper angles (greater than 60º), you may want to use a CBR 2™ motion detector clamp (sold separately).

See pages 6–9 for hints on effective data collection.

Typical plots

15¡30¡

Typical answers

1.the third plot

2.time; seconds; distance of object from CBR 2™ motion detector; feet or meters

3.varies (should be half of a parabola, concave up)

4.a parabola (quadratic)

5.varies

6.varies (should be parabolic with increasing curvature)

28 GETTING STARTED WITH THE CBR 2™ SONIC MOTION DETECTOR

Notes for Teachers

7.0¡ is flat (ball can’t roll); 90¡ is the same as a free-falling (dropping) ball

Explorations

The motion of a body acted upon only by gravity is a popular topic in a study of physical sciences. Such motion is typically expressed by a particular form of the quadratic equation,

s= ½at2 + vit + si where

0s is the position of an object at time t

0a is its acceleration

0vi is its initial velocity

0si is its initial position

In the quadratic equation y = ax2 + bx + c,

yrepresents the distance from the CBR 2™ motion detector to the ball at time x if the ball’s initial position was c, initial velocity was b, and acceleration is 2a.

Advanced explorations:

Since the ball is at rest when released, b should approach zero for each trial. c should approach the initial distance, 0.5 meters (1.5 feet). a increases as the angle of inclination increases.

If students model the equation y = ax2 + bx + c manually, you may need to provide hints for the values of b and c. You may also direct them to perform a quadratic regression on lists L1, L2 using their calculators. The ball’s acceleration is due to the earth’s gravity. So the more the ramp points down (the greater the angle of inclination), the greater the value of a. Maximum a occurs for

q= 90¡, minimum for q = 0¡. In fact, a is proportional to the sine of q.

© 1997, 2004, 2006 TEXAS INSTRUMENTS INCORPORATED

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Texas Instruments CBR 2 manual Activity 5-Rolling Ball, Concepts
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CBR 2 specifications

Texas Instruments has long been a prominent player in the field of educational technology, and the CBR 2 (Calculator-Based Ranger 2) is a testament to their commitment to enhancing the learning experience, particularly in the realms of mathematics and science. Designed to complement graphing calculators, the CBR 2 is a versatile data-collection device that empowers students and educators to explore real-world phenomena through hands-on experimentation.

One of the main features of the CBR 2 is its ability to capture a wide array of data through various sensors. The device is equipped with an array of built-in sensors that can measure motion, including speed and distance. This makes it an invaluable tool for physics experiments, allowing students to visualize concepts such as speed, acceleration, and trajectory.

The CBR 2 utilizes ultrasonic technology to detect distance through sound waves. This feature enables students to conduct experiments that demonstrate principles of sound and motion in a tangible way. With a range of up to 6 meters, the CBR 2 provides accurate and reliable measurements that can be graphically represented using compatible Texas Instruments graphing calculators.

The device is highly user-friendly, with simple interfaces that allow users to easily collect and analyze data. The integration with graphing calculators simplifies the process of data visualization, enabling students to create graphs in real time as they conduct experiments. This capability is particularly beneficial in encouraging interactive learning and fostering a deeper understanding of scientific principles.

The CBR 2 is designed to be portable and durable, making it suitable for classroom settings as well as outdoor experiments. Its compact size and lightweight construction ensure that it can be easily transported, allowing educators to take learning beyond the confines of the classroom.

The CBR 2 also supports various modes of data collection, including Event Mode, which allows users to trigger data collection based on specific events. This feature is useful in demonstrating concepts such as projectile motion and collisions, providing students with hands-on experience that enhances their learning.

In summary, Texas Instruments' CBR 2 is a powerful educational tool that enables students to collect, analyze, and visualize data in an engaging manner. With its built-in sensors, ultrasonic technology, and seamless integration with graphing calculators, the CBR 2 stands out as a versatile device that enriches the educational experience. It not only provides a platform for conducting experiments but also cultivates critical thinking skills and a deeper understanding of scientific concepts, preparing students for a future in STEM fields.
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