We're going to calculate how long it takes for rain to refill the Earth's oceans. We might want to know this number if, for instance, we were comparing Earth's oceans and Mars', or if we were looking at how long it would take water pollution on Earth to cycle its way through the entire water system (ocean -> cloud -> rain -> river -> ocean).
We can solve this problem (beyond guessing) by breaking it up into little pieces, each of which is simple enough by itself. We can then put the little pieces together and solve it the same way we calculate the number of sand grains on the Earth's beaches in class.
a. First, just for comparison, take a random guess at how many drops of rain fall in a year: ___________ drops. And, how long would it take for raindrops to fill up the Earth's entire oceans: __________ years. Don't worry about being accurate: just guess, and you can compare it to the number you'll calculate later on.
b. Let's start by calculating the amount of rain that falls on the
earth every year. A good place to start might be assuming it's about a
layer 1 meter deep - this might be wrong, but the amount of rain is
probably more than a centimeter, and less than 10 meters, so this is a
good place to start. (Feel free to pick your own value, too.) We can
multiply the area of the Earth (call it a sphere of radius 7000 km) by
the depth of the rain, and get the total volume of the rain that falls
every year:
Surface area of the Earth (written A_earth) = ____________ cm2.
Volume of rain every year (written V_year) = _________ cm3/yr.
Convert units into seconds: V_second = ___________ cm3/s.
(You'll probably want to convert the units into cm before doing the math.)
c. How big is an average raindrop: 1 mm across? 3 mm? 1 cm? Pick
a value, and assume the raindrop is a sphere and calculate its volume:
V_drop = _________ cm3.
Now, divide the total volume
from above (V_year) by the volume of each raindrop, to calculate the
total number of raindrops that fall each year. Be sure your units are
right before dividing, since you can't divide mm3 by
km3 and get a meaningful number! It's probably easiest to
do everything in cm.
V_year / V_drop = _______________ rain drops in the ocean.
d. What is the volume of the Earth's oceans? We can calculate this
based on assuming they cover perhaps a layer over the entire surface
about 10 km deep. Again, this number isn't 100% accurate, but the real
value is somewhere between 1 km and 100 km, so 10 km is a good guess -
or look up a better number. Of course, oceans don't cover completely
all of the Earth (only about 70%), but we'll assume for now that they
do. Using the same technique as in b), calculate the volume of the
oceans:
V_ocean = _______________ cm3.
e. Finally, let's divide the volume of the ocean by the rate at
which rain falls, and see how long it takes for rain to fill the
oceans:
V_ocean / V_year = __________ yr.
f. Does this number surprise you? Does it seem particularly long or short? How does it compare to what you guessed? Are there major assumptions that we made that are incorrect? Do you think the `real' value is larger or smaller than what we calculated here?
b) What would be the delay time of a rover on the Moon, and could it be controlled directly by a person on Earth?
Let's say we sent rover to a planet in the Orion nebula, 450 ly away:
c) If our spacecraft traveled to Orion at one tenth the speed of light (written
as 0.1 c), how long would it take for it to reach Orion?
Hint: this is easier than it looks! How long would it take to go 450
ly if our spacecraft went at the speed of light? Half the
speed of light? Ten times the speed of light?
d) How long would it take for radio signals from Earth to reach Orion and return to us? Would this be a practical way to explore nearby planetary systems?
Last modified 6-Jun-2000