# Relativity in Lesson Plans

This June, the department conducted a summer workshop for 10 physics teachers on experimental evidence for Einstein's special relativity. As part of their work, the teachers presented lesson plans that require students to notice that different observers will report different measurements of identical events. The following is a summary of those ideas.

• ##### The Doppler effect for the size of the universe.
Students observe the change in pitch of the sound of a buzzer whirling in a circle or passing on a bicycle, or flying by attached to a ball. The pitch change can be used to determine the speed of the buzzer relative to the observer. This idea is carried over to the Doppler effect for light and its use as a (relative) speedometer for stars. Hubble's correlation between a star's recession speed and its distance from us completes the connection to the size of the universe.
• ##### Doppler effect for finding the speed of a car.
A car drives by a tape recorder which records the sound of the car's horn as it passes. The car drives fast enough to produce a noticeable Doppler shift. The sound is transferred to a computer where its spectrum in analyzed to find the size of the shift in pitch. (One program well suited to this analysis is CoolEdit by Syntrillium Software, for Windows machines. This program will determine the spectrum of small segments of the tape recording.)
• ##### Triangulation measurement of the height of a rocket flight.
Rockets made from 2 liter bottles are observed by three observers; the student who launches the rocket and must look nearly straight up to see it, and two students 30 meters away who see the top of the flight at an angle of perhaps 45 degrees above the horizon. The launching student sees an angle above the horizon of nearly 90 degrees, no matter what the height. The distant students measure angles that vary strongly with the height. Comparing uncertainties shows that some observation points are better than others (a point often missed by those who like to say "It's all relative.")
• ##### Velocity measurement from a "moving" platform (two embodiments).
1. The speed of a "target" rolling cart is measured by two ultrasonic position detectors: One fixed in the laboratory, and one attached to a second, powered, cart. The powered cart follows the target cart, but does not keep up with it. A third ultrasonic position detector measures the velocity of the powered card in the laboratory frame of reference. The two laboratory frame measurements are used to predict the target's velocity that will be observed by the powered cart.
2. An ultrasonic detector measures the velocity of a falling ball. The detector sits above the ball and "looks" down. The detector is mounted on one mass of a (balanced) Atwood machine so that it can be moved (at constant rate) either upward or downward. The velocity of the Atwood masses is measured separately and used to calculate the lab velocity of the ball (as a function of time) from the moving ultrasonic detector data.
• ##### Analysis of muon data.
Muon measurements were made during the 1-week workshop, concluding with a half-day measurement at higher altitude, at the Burnt Cabins Pennsylvania Turnpike Maintenance Center. (The photo shows experiments in relative motion while the experiment was running.) Measurements continued on a longer time scale, and are being reported in a separate web site available to the participants. In the lesson, students will download and analyze the data for evidence of Einstein's time dilation.
• ##### Dropped ball trajectories.
Four meter sticks are laid end to end to create a one dimensional coordinate system. A student walks rapidly along the meter sticks and drops a ball. The trajectory is analyzed in terms of the position of the ball when dropped and the point at which it hits the floor. The experiment is repeated with the meter sticks set into motion, both "with" and opposite to the walking student's velocity. The results are analyzed in terms of the relative motion of student and meter stick. (A concrete connection is made by raising the question of why we bother to make divided highways.)
• ##### Thrown ball trajectories.
A ball is launched upwards from a platform which can be stationary in the laboratory, and can also move in the laboratory. Students describe quantitatively and qualitatively the path observed by a person fixed in the laboratory. The parabolic shape that is observed when the platform moves relative to the laboratory is also evidence that justifies separating motion into horizontal and vertical components.
• ##### The Coriolis effect.

This effect arises when an object moves from one "natural" frame of reference to another "natural" frame of reference. What is "natural" is determined by the local geography. On the earth surface the two frames of reference typically are fixed to the earth at two different latitudes, so that they have different tangential velocities as they rotate with the earth about its axis. When the northern observer reports the horizontal component of velocity of a ball headed south, the southern observer will conclude that a force was necessary to account for the (changed) horizontal component of velocity observed in the southern frame of reference. (We assume that the ball velocity as measured in an inertial frame of reference is unchanging, except for acceleration in the vertical direction.)

In the lesson plan, the changes in observer velocity are more extreme. Two students sit opposite to each other, across a diameter of a playground merry-go-round. For concreteness, assume that the merry-go-round spins so that each student is moving to his/her right. In the playground frame of reference their velocities are of equal magnitude,V, but opposite direction, as the merry-go-round spins. Student A throws a ball to student B, aiming it so that it should pass over the center of the merry-go-round. Student B is surprised at how hard it is to catch the ball.

Student B observes the ball coming with an additional velocity component, in the direction of A's tangential velocity. To account for the difference between B's and A's reported velocity for the same ball we find in convenient to say that the ball suffered an acceleration while it changed location from near A to near B. Following the Newtonian logic, we say that the acceleration was caused by a new force, the Coriolis force.