Two race cars are parked in single file on an
abandoned highway along which the cars' drivers plan to drive very fast.
The cars are parked 333 meters (m) apart, which is the distance that sound
travels through the cool air in 1 second (s). The drivers plan to
maintain this safe distance between the cars by beeping their horns to send sound signals to
one another. Whenever the driver of the front car beeps the horn, the driver of the rear car
replies with a return beep. If the return beep is received at the front car 2 s after the
initial beep is sent, the driver knows that the cars are spaced correctly. If the round-trip
signal time is not 2 s, the driver adjusts the distance to 2 s.
Time aboard each car is kept by a "sonic clock" which consists of
two sound-emitting-and-detecting units, U and U', mounted 3.33 meters apart at the tips of
two antennae atop the car (as shown in the front view of car at right). A clock keeps time by
sending a very brief sound signal back and forth between U and U'. When a signal from one unit
is detected by the other unit, a return signal is immediately emitted. In this way a signal is
constantly traveling back and forth between U and U'. When a car is not moving, the signal
makes 50 round trips between U and U' every second. And for every 50 round trips, the car's
time display on the instrument panel automatically advances one second. |
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As the cars gradually accelerate, the driver of the front car
periodically checks the separation distance and maintains the 2 s round-trip signal time
according to the sonic clock on the instrument panel. This results in a decrease in the
333 m separation distance. When the speed of the cars reaches Mach .5 or 166 m/s,
the separation distance is .86 times the 333 m at-rest separation or
288 m.1
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Speed: Mach 0 = 0 m/s Length of 2-car system: 333 m |
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Speed: Mach .5 = 166 m/s Length of 2-car system: 288 m |
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If the speed of the 2-car system increases to Mach .8 or 266 m/s, the separation distance
decreases to .6 times the at-rest separation, or 200 m. The ratio, r, of the separation
distance in the moving system to the at-rest separation is a function of the velocity
Vm as follows, where Vm is
the velocity of the 2-car system through the air in Mach number units.
As the speed of the 2-car system approaches the speed of sound,
the separation distance approaches zero because the speed of the sound energy moving from the
rear car toward the front car approaches zero relative to the cars. This contraction
effect and slowing of time in the 2-car system moving through the medium through which sound energy
is propagated helps understand similar phenomena discussed in the quantum medium view of this web
site. The analogy shows primary and secondary effects of the system's motion through the medium.
The motion causes a decrease in the rate of round-trip energy exchange between the subsystems of
the system, and this in turn results in a change in the system's length and the standard of
time in the system. For any velocity Vm of the
2-car system, the ratio of the rate of round-trip sound energy exchange in the system to the
at-rest rate is equal to r in the above equation. As a result, r is also the ratio of the rate
of the sonic clocks in the moving system to their at-rest rate.2
In the quantum medium view of this web site, it is
electromagnetic energy, in the form of photons, which is propagated through a quantum medium
and is exchanged between the constituents of atoms. It is the rate of exchange of this energy
that determines the spatial relationships within and between atoms. The motion of any system
of atoms through the medium affects the pattern of energy exchange and the rate of round-trip
energy exchange in the system with consequences that are analogous to, but much more profound
than, those of this analogy.
1. The Mach number is the ratio of a system's speed through the air to the speed of sound
through the air.
2. For example, assume that the 2-car system has a velocity of Mach .8 which results in r=.6
in the equation above. In this case the velocity of sound between U and U' is Mach .6 because
the sound energy moving between U and U' must have a component of its Mach 1 velocity through
the air be Mach .8 in the direction of motion of the cars. Because the sound energy moving
between U and U' is moving at only .6 times its at-rest rate, the rate of the sonic clocks
is only .6 times their at-rest rate. Therefore, 2 seconds on the sonic clocks is 2/.6 or 3.333
seconds on an at-rest clock, which is the time duration for the round-trip beeping horn signal
between the cars. The time for a horn beep to travel from the front car to the rear car is
the 200 m separation distance divided by Mach 1.8 or 599.4 m/s, which is .333
seconds. And the time for a horn beep to travel from the rear car to the front car is
200 m divided by Mach .2 or 66.6 m/s, which is 3 seconds. Therefore, the round-trip
travel time for the sound energy is 3.333 seconds or the 2 seconds shown on the sonic clocks.
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Analogy which helps understand the quantum medium view.