NASA
satellite orbits the earth every 90 minutes. During anorbit,
the
satellite's electric power comes either from solar array
wings,
when these are illuminated by the sun, or from batteries.
The
batteries discharge whenever the satellite uses more
electricity
than the solar array can provide or whenever the satelliteis
in
the shadow of the earth (where the solar array cannot beused).
If
the batteries are overused, however, they can be
damaged.
"font-family:'comic sans ms',
sans-serif;font-size:medium;">You are
to determine whether the batteries could be damaged ineither
of
the following operations. You are told that the batterycapacity
is
50 ampere-hours. If the total battery discharge does notexceed
40%
of battery capacity, the batteries will not be
damaged.
(a)
Operation 1 is performed by the satellite while orbiting the
earth.
At the beginning of a given 90-minute orbit, the satellite
performs
a 15-minute maneuver which requires more current than the
solar
array can deliver, causing the batteries to discharge. The
maneuver
causes a sinusoidally varying battery discharge of period 30
minutes with a maximum discharge of ten amperes at 7.5minutes.
For
the next 45 minutes the solar array meets the total
satellite
current demand, and the batteries do not discharge. Duringthe
last
30 minutes, the satellite is in the shadow of the earth and
the
batteries supply the total current demand of 30
amperes.
(i)
The battery current in amperes is a function of time. Plot
the
function, showing the current in amperes as a function oftime
for
the 90-minute orbit. Write a formula (or formulas) for the
battery
current function.
(ii)
Calculate the total battery discharge (in units of
ampere-hours)
for the 90-minute orbit for Operation 1.
(iii)
What is your recommendation regarding the advisability of
Operation
1?
"font-family:'comic sans ms',
sans-serif;font-size:medium;">(b)
Operation 2 is simulated at NASA's laboratory in Houston.
The
following graph was produced by the laboratory simulation of
the
current demands on the battery during the 90-minute orbit
required
for Operation 2.
(i)
Calculate the total battery discharge (in units of
ampere-hours)
for the 90-minute orbit for Operation 2.
(ii)
What is your recommendation regarding the advisability of
Operation
2?