The Mathematics of Video Tapes
Louis Nemzer
The advent of the videocassette has
allowed not only the conservation and propagation of actual events, but also
imaginary images. This medium has made it possible to record and distribute
visual information in a whole new way. Moreover, despite its importance, its
inner workings are exquisitely simple, allowing videotapes to be made
inexpensively. Paradoxically, this simplicity gives most electromagnetic media like
videotapes (plus audio and computer tapes) a special place in mathematics not
shared by movie reels.
The main difference between movie reels and videos is that movie reels
are composed of actual pictures, while videos are made of magnetic tape. Consequently,
movie reel projectors must run at a constant speed. However, if the movie was
played by pulling one reel with a constant angular velocity, the linear speed
of the reel would vary depending on how much of the reel had already been
pulled.
The reason is that as the movie goes
on, the take-up reel would still takes the same amount of time to go around
once, but since the circumference would be larger, the tape pulled each time
around would have be greater than during the beginning of the movie. Therefore,
movie reels are NOT pulled by the reel; rather, they have holes in the sides so
that the reel called sprockets that allow the reel to be pulled at a constant
linear rate.
However, videos do not have this problem. It is all right to
record each second at a different linear speed, thus "stretching" the
signal as you go along, as long as each section is played at the same speed it
was recorded. This is not a problem because the radius of the take-up reel at
any point, and therefore the linear speed, will be the same each time each time
you play the video.
The only drawbacks with the video
tape system are: First, that splicing sections of video tape from one position
to another or from one tape to another will most likely cause the signal to be
noticeably distorted. (This is another reason why movies are made differently;
it is often necessary to splice frames in a movie). Second, video tapes cannot
be made to long because the amount of tape needed for each second increases the
longer the tape gets (That's why you don't see many video cassettes that run
longer than six hours). Finally, your VCR counter, which gages how much tape
had passed, will not run at a constant speed.
(The one advantage video tapes get
is the fact that since they can record "stretched" signals, you can
record different amounts of footage per area, depending on the picture quality
needed and time constraints. For example, EP setting, the slowest speed, gives
you 6 hours of recording time on a normal video but the picture quality is
slightly compromised. SP, on the other hand, gives you the best picture quality
but only 2 hours of taping. LP, in the middle, provides 4 hours of footage and
a corresponding picture quality.)
Despite these complications,
calculus makes it possible for us to relate the time t to the amount of tape
that has already been pulled. Therefore, the question of this report is:
Find a general equation for s, the total
amount of tape already pulled to the take-up reel of in a video tape at time T,
in terms of T; W, the constant angular velocity of the take-up reel; C, the
radius of the plastic core of the take-up reel; and Z, the thickness of the
tape.
SOLUTION
We know that the radius of the tape
will be C + Z (the number of revolutions so far), so if we define r the radius
of the tape, we can say that at any time t:
R = C + Z (WT / 2p)
Or:
(1) R =
(ZW / 2p) T + C
Since the amount of tape that will
be pulled in any time interval is equal to the arc length of the tape swept out
by the amount rotated in that interval, q. Since r grows very slowly, we can hold it constant for
short time intervals and use a linear approximation to say that for a small dT:
dS
= R dq
By
the definition of a differential, dS equals (dS/dq) dq, we see that:
(2) R =
(dS/dq)
By using the chain rule, (dy/dx) =
(dy/du) (du/dx), we can say that:
(3) (dS/dT)
= (dS/dq) (dq/dT)
If we substitute R for (dS/dq) [equation 2] and W for (dq/dt) [definition of angular velocity] we get:
(dS/dT)
= RW
Substituting for R from equation 1:
(dS/dT)
= [(ZW / 2p)T + C] W
Or:
(4) (dS/dT)
= (ZW2 / 2p)T + CW
Integrating with respect to t gives
us:
S = (ZW2 / 4p)T2 + CWT + Q
Where Q is the arbitrary constant.
But since S = 0 when T = 0, Q must also equal zero. Therefore the equation for
S is:
S = (ZW2 / 4p) T2 + CWT
(Note
that after equation 4, the solution became a simple initial value problem)
In conclusion, it indeed seems that
the simplicity of videotapes and other electronic media makes them good topics
for calculus projects.