You are a passenger in a car that is driving down the highway at 100 km/h (60 mph). You are sitting in the right-rear seat and all the windows are closed tightly. The air flowing around the car makes a sharp inward bend as it travels around the windshield and past the right-front window (the front window on the passenger-side of the car). By the time this air flows past your window, it has straightened out and is traveling almost directly backward toward the rear of the car.
1. When your friend in the front passenger seat opens his window a little, your ears pop due to a drop in the car's air pressure. However, when you then open your window a little, the pressure in the car rises back toward normal atmospheric pressure. Explain why (a) opening the front passenger-side window caused a sudden decrease in the car's internal air pressure, and (b) why opening your window caused that pressure to increase back toward atmospheric pressure.
Answer: (a) Because of the strong inward bend near
the front window, the pressure at the window's surface is low and opening the
window allows car air to accelerate toward that low-pressure part of the
airstream. (b) Since air is going straight near the back window, the pressure
at the window's surface is essentially atmospheric and opening the window
allows this air to accelerate toward the low pressure inside the car.
Why: The bending
airstream creates a low pressure just outside the front window. When you crack
that window open, you don't change the pattern of air much, but you do allow car
air to accelerate toward that window and the low pressure outside it. As air
leaves the car, the air pressure inside the car drops. Opening the rear window
allows atmospheric pressure air outside that window to flow into the car and
replace the missing air molecules. The car's air pressure rises back toward
normal atmospheric pressure.
2. With both windows slightly opened, in what way is air circulating through the car? Use this result to explain why it is that when you friend throws his gum wrapper out the front-right window, it comes back inside through the back-right window and hits you in the face. (Good thing it was just the wrapper!)
Answer: Air is flowing out of the car through its
front window and back into the car through its rear window. This air
circulation carries the gum wrapper with it so that after it leaves the front
window, it reenters the car through the back window.
Why: Air flowing steadily
out of one window must eventually be replaced by air flowing in through another
window. In a car, the lowest pressure portion of the airstream is outside the
front side window, so that is where air normally leaves the car. It then
reenters the car through any other opened window. Passengers in the rear seats
are often hit by debris thrown out the window by people in the front seats.
It's a self-enforcing anti-littering program.
3. You now open your window completely and stick your right hand out into the passing stream of air. With your palm facing directly forward, into the onrushing wind, you feel a tremendous force pushing your hand toward the rear of the car. What is this force, and why does it occur?
Answer: The force is pressure drag and it occurs
because the air slows down in front of your hand, creating high pressure there,
but doesn't slow down behind your hand. (Equivalently, your hand creates a
large turbulent wake and transfers some of its forward momentum to that wake.)
Why: The air pressure in
front of your hand is much higher than the air pressure behind it. That's
because the air slows down nicely in front of your hand, losing kinetic energy
and gaining pressure potential energy, but it doesn't slow down again behind
your hand. Instead, it detaches from the surface of your hand and forms a large
turbulent wake of roughly atmospheric pressure air. With more pressure in front
than behind, your hand experiences a net backward force; the force of pressure
drag.
4. You now tilt your hand so that your palm is facing forward and somewhat downward. You feel a new force that makes your hand feel lighter and it almost floats upward of its own accord. What is this force you feel, and what causes it?
Answer: The force is lift and it occurs because you
are deflecting the airstream downward and it is pushing upward on your hand in
response (equivalently, you are creating an inward bend above your hand and an
outward bend below it, so that the air pressure above your hand is lower than
the air pressure below it and the pressure imbalance pushes your hand upward).
Why: Your hand is flying
through the air. When you tilt it properly, it behaves like a simple airfoil
and obtains lift from the passing airstream.
Traveling to mars is a challenge, in part because getting there takes so long. If we could speed up the space ship, travel time would be less of a problem. One possibility is the VASIMR engine, an engine that uses radio waves to heat its exhaust so hot that the exhaust leaves the engine at 300 km/s (about 700,000 mph). A normal rocket exhaust speed is only 4 km/s, so the VASIMR engine exhaust travels about 75 times as fast as normal engine exhaust.
5. Explain why each kilogram of exhaust ejected at 300 km/s provides 75 times as much thrust as each kilogram of exhaust ejected at 4 km/s. (We are assuming a specific amount of fuel use each second.)
Answer: At the faster exhaust speed, each kilogram
of fuel carries away 75 times as much momentum as it would have at the slower
exhaust speed (momentum is proportion to velocity). Since fuel is being used at
a steady rate, the rocket receives 75 times as much impulse (momentum transfer)
each second. Since impulse is proportional to the force that causes it, the
force (thrust) on the rocket by the fuel must increase by a factor of 75.
Why: Rocket engines are
rated according to specific impulse: how much momentum they transfer to the
rocket in the process of ejecting their exhaust gases. This impulse is the
product of how hard the engine pushes (the thrust) times how long the engine
pushes (the burn time). In this question, the burn time isn't changing, but the
specific impulse is going up by a factor of 75. That means that the thrust must
also go up by a factor of 75.
6. Calculating exactly how fast a rocket will travel after ejecting all of its fuel as exhaust is complicated by the fact that the rocket's speed changes while its engine operates. For simplicity, suppose that a rocket could eject all of its fuel at once, while the rocket is still at rest. Now suppose that the total mass of the rocket consists of 1 part spaceship and 10 parts of fuel. Explain why the spaceship portion of the VASIMR rocket would reach a final speed of 3,000 km/s.
Answer: The rocket starts with zero total momentum
so its parts must end up with zero total momentum. The fuel portion, with 10
parts of mass, and the ship portion, with 1 part of mass, must each end up with
same amount of momentum in opposite directions. Since the fuel ends up with a
speed of 300 km/s, the ship must end up with a speed of 3,000 km/s in the
opposite direction in order for their momenta to sum to zero.
When the two
"objects" separate, ship and fuel, they continue to have zero total
momentum. Since the ship has only one-tenth the mass of the fuel, it must have
ten times the speed of the fuel. That way, they both have the same amount of
momentum but in opposite directions.
7. The VASIMR-powered rocket must be careful not to eject its fuel too quickly because that would injure the astronauts on board. Why would sending out too much fuel each second at the 300 km/s exhaust speed of the VASIMR engine injure the astronauts? (Note that this question has nothing to do with radiation sickness problems or anything like that.)
Answer: Sending out too much fuel each second will
produce too much thrust. The ship will accelerate so quickly that the forces
exerted on the astronauts to keep them accelerating with the ship will injure
the astronauts.
Why: At more than a few
times the acceleration due to gravity (a few g's), the forces required to keep
an astronaut accelerating along with the ship will cause internal damage to the
astronauts. Furthermore, the astronauts will pass out because their circulatory
systems won't be able to keep the blood moving properly in the direction of the
acceleration.
8. Getting to mars is one thing; stopping for a visit and then returning to earth is another. If the rocket needed to be 10 parts fuel to 1 part rocket in order to move quickly enough to travel past mars in a short amount of time, what would it have to be (in terms of parts fuel to parts rocket) to reach mars in a short amount of time, stop for a visit, and then return to the earth in a similar short amount of time, and stop here?
Answer: The rocket would have to be about 10,000
parts fuel to 1 part rocket. (Actually, 14,641 parts fuel to 1 part rocket, but
that's unnecessarily exactly.)
Why: To get up to speed
on its way to mars, the rocket must eject all but about 10% of it mass as
exhaust. Stopping the rocket upon arrival at mars is just as difficult as
starting, so the rocket must eject all but about 10% of its remaining mass as
exhaust. To get up to speed on its return, it must again eject all but about 10%
of its remaining mass as exhaust. And finally, it must eject all but about 10%
of its remaining mass as exhaust to come to a stop at the earth. The overall
ship portion of the original rocket is 10% of 10% of 10% of 10% or 0.01% of the
original rocket. The original rocket must be about 99.99% fuel and 0.01% ship.
Not likely. The more exact fraction comes from the fact that each step requires
that all but 9.0909% of the remaining mass be ejected as fuel. Taking that
percentage 4 times gives only 0.00683% or, even more exactly, 1/14641.