Download Understanding Gravity: The Force that Shapes Our Universe and more Summaries Physics in PDF only on Docsity!
3. Force and Gravity
Being in orbit is like being infatuated – you are constantly falling, but you aren’t getting closer.
The Force of Gravity
Any two objects that have mass attract each other with a force we call gravity. You probably never noticed this for small objects, because the force is so weak. But the Earth has lots of mass, and so it exerts a big gravitational force on you. We call that force your weight. The fact that gravity is actually a force of attraction is not obvious. Prior to the work of Isaac Newton, it was assumed that gravity was simply the natural tendency of objects to move downward.
If you weigh 150 lb, and are sitting about 1 meter (3.3 feet) from another person of similar weight, then the gravitational force of attraction between the two of you is 10-7^ lb. This seems small, but such forces can be measured; it is about the same as the weight of a flea.
You weigh less when you stand on the Moon, because the force of attraction is less. If you weigh 150 lb on the Earth, you would weigh only 25 lb on the Moon. You haven’t changed (you are made up of the same atoms), but the force exerted on you is different. Physicists like to say that your mass hasn’t changed, only your weight. Think of mass as the amount of material, and weight as the force of attraction of the Earth (or whatever other planet or satellite you are standing on).
Mass is commonly measured in kilograms. If you put a kilogram of material on the surface of the Earth, the pull of gravity will be a force of 2.2 lbs. So a good definition of a kilogram is an amount of material that weighs 2.2 lbs when placed on the surface of the Earth. That number is worth remembering.^1 Go to the surface of Jupiter, and you will weigh nearly 400 lbs. On the surface of the Sun you will weigh about 2 tons, at least for the brief moment before you are fried to a crisp. But in all cases your mass will be 68 kg.
The equation that describes the pull of gravity between two objects was discovered by Isaac Newton. It says that the force of attraction is proportional to the mass – double the mass and the force doubles. The force also depends on the distance. It is an inverse square law. It is inverse because when the distance gets larger, the force gets smaller. It is a square law because if you triple the distance, the force decreases by nine; if you make the distance increase by 4, then the force goes down by 16, etc.
(^1) A more accurate value is that there are 2.205 lb in a kilogram, and 0.4536 kg in a
pound, but don’t bother memorizing these more precise numbers.
The Equation of Newton’s Law of Gravity
“Newton’s Law of Gravity” gives the gravity force between two objects with masses m and M separated by distance r :
F G
m M r^2
G is called the Gravitational Constant, and has the value
(^) 6.67 10 ^11 N m^2 kg ^2 (N is for Newton , the physicists’ unit of force) or
1.5 10 ^11 lb m^2 kg ^2. Let’s go back to the example given in the text: two 150 lb people separated by 1 meter. The mass of each person is
150 lb 2.2 lbkg
68 kg. Putting these into the formula gives
F 1.5 10 ^11 lb m^2 kg ^2 (68 kg )
2 (1 m )^2
10 ^7 lb.
Newton’s Law of Gravity actually gives the force only between two small objects. If one of the objects is a sphere (such as the Earth) then it turns out that you can still use the formula, but you must use the distance to the center of the sphere as the value for r. As an example, let’s put in numbers for a 1 kg object sitting on the surface of the Earth. Then the force of attraction is given by the gravity equation with m = 1 kg, M = the mass of the Earth = 6 1024 kg, and r = radius of the Earth (that’s the distance to the center of the sphere). This distance is r = 6371 km ≈ 6 106 meters. Without plugging in the numbers, can you guess what the answer will turn out to be? Guess, and then check this footnote^2 to see if you guessed correctly.
Suppose you weigh 150 lbs on the Earth. Then your mass is
150 lb 2.2 (^) kglb
68 kg. What will
you weigh on the Moon? We can calculate that by using Newton’s Law of Gravity, and putting in the M = the mass of the Moon = 7. 3 1022 kg, r = the radius of the moon =
- 7 106 meters. The answer is F = 25 lb. That means you will weigh 25 lb. on the surface of the Moon.
Newton’s Third Law
Here is something that might surprise you: if you weigh 150 lb, not only is the Earth attracting you with a force of 150 lb, but you are attracting the Earth with a force of 150 lb too. This is an example of something called “Newton’s third law” – if an object exerts a force on you, then you exert the same force back on it.
(^2) The answer is 2.2 lb. Of course, that is the weight of a 1 kg object.
Also, 1 newton ~ 4 ½ pounds, so we could express the answer as about ½ newton.
fell together. You would look just like the astronauts floating around in the Space Station.
Now imagine that instead of falling, you and the elevator are shot together out of a gun, and fly 100 miles before hitting the ground. During that trip, you will again feel weightless. That’s because you are in motion along with the elevator. You and it fly in the same arc.^4 Your head and chest are both moving in that arc together; there is no force between them and your neck muscles can be completely relaxed. Your head will seem to have no weight. Prior to the space program, potential astronauts were flown in airplanes following such arcs in order to see how they responded to the sensation of weightlessness
- and to get them used to it.
When you and the elevator are moving together under the force of gravity (either falling or shot in an arc) there seems to be no gravity. From that alone, you might think you were far out in space, far away from the gravity of any planet, star, or moon. From inside the elevator, you can’t tell the difference.
Now imagine that at the top of a very tall tower (200 km high) is a large gun, pointing horizontally. We shoot the elevator and you horizontally. If we picked a low velocity (e.g. 2 km/sec) you and the elevator would curve towards the Earth, and you would crash into it, as in path A in the figure below. But instead we pick a high velocity: 8 km/sec. You and the elevator are shot from the gun, and you curve towards the Earth, but because of your high velocity, you miss the edge of the Earth, as in path B. You keep on curving downward, but you never hit. You are in orbit. The force of gravity makes the path of the elevator – let’s call it a space capsule now – curve downwards. But if that curvature matches the curvature of the Earth, then it misses the surface, and stays at a constant height.^5
Figure: Capsule shot into space from a tower
(^4) Your path is a geometric curve known as a parabola.
(^5) If your velocity is not exactly horizontal, or if your velocity is a little low or high, then
the orbit will not be a circle but an ellipse.
This may seem preposterous, but it is reasonable to think of an astronaut in orbit around the Earth as being in a state of perpetual falling. That’s why he feels weightless.
You can think of the Moon as doing the same thing. It is attracted to the Earth by gravity, but it has high sideways motion. Even though it is falling towards the Earth, is always misses.
The Velocity for Low Earth Orbit (LEO)
To stay in a circular orbit just a few hundred miles above the surface, the velocity of the satellite must be about 8 km per second, which is about 18,000 miles per hour. (The actual value depends slightly on the altitude; we’ll derive this number from a calculation later in this chapter.) At this velocity, the satellite orbits the 24,000 miles circumference of the Earth in about 1.5 hours.
If the astronaut wants to land, he does not point his rockets downward and fire his rockets away from the Earth; he fires his rockets towards the direction he is headed – in the forward direction! That slows down the satellite, so it is no longer going fast enough to miss the edge of the Earth. Gravity brings the satellite back to Earth. If the satellite moves faster than 8 km per second, it leaves the circular orbit and heads out into space. At about 11 km per second it will have sufficient velocity to reach the Moon and beyond. This velocity is called the escape velocity. We’ll discuss this concept further later in this chapter.
Analogy with a Rock and Sling
There is another way to think about Earth satellites. Forget gravity for a moment. Imagine that you have a rock tied at the end of a string, and you are spinning it in a circle above your head. The string provides the force that keeps the rock from flying away, that keeps the rock in circular motion. If the string breaks, the rock flies off in a straight line. Gravity does the same thing for an Earth satellite: it provides the force that keeps the satellite in a circular orbit.
An old weapon called the “sling” is based on this principle. A rock is held by a leather strap, and spun in circles over the head. Arm motion helps it pick up circular speed. It is the strap that keeps the rock in circular motion. When the strap is released, the rock flies in a straight line towards its target. Such a sling was the weapon that, according to the Bible, David used to kill the giant Goliath.
In a similar manner, if we could suddenly “turn off” the force of gravity, the Moon would leave its circular orbit, and head off in a straight line. Likewise for all the satellites in orbit around the Earth. And with the Sun’s gravity turned off, the Earth would head out into space too, at its previous orbital speed of 30 km/sec (67,500 mph).
Spy Satellites
Spy satellites are satellites that carry telescopes to look down on the surface of the Earth and see what is going on. They were once used exclusively by the military, to try to see the secrets of adversaries, but now they are widely used by government and industry to look at everything from flooding and fires to the health of food crops.
The ideal spy satellite would stay above the same location all the time. But to do that, it must be geosynchronous, and that means that its altitude is 22,000 miles. At those distances, even the best telescopes can’t see things smaller than about 200 meters. (We’ll derive that number when we discuss light.) That means that such a spy satellite could see a football stadium, but couldn’t tell if a game was being played. Such satellites are good enough to watch hurricanes and other weather phenomena, but are not useful for fine details, such as finding a particular terrorist.
Thus, to be useful, spy satellites must be much closer to the Earth. That means they must be in low earth orbit (LEO), no more than a few hundred miles above the surface. But if they are in LEO, then they are not geosynchronous. In LEO they zip around the 24, miles circumference of the Earth in 1.5 hours; that gives them a velocity relative to the surface of 16,000 miles per hour. At this velocity, they will be above a particular location (within ± 100 miles of it) for only about 7.5 minutes.^7 This is a very short time to spy. In fact, many countries that want to hide secret operations from the United States keep track of the positions of our spy satellites, and make sure their operations are covered over or hidden during the brief times that the spy satellite is close enough to take a photo.
LEO satellites cannot hover. If they lose their velocity, they fall to Earth. If you want to have continuous coverage of a particular location, you must use a circling airplane, balloon, or something else that can stay close to one location.
GPS – Medium Earth Orbit Satellites (MEO)
One of the wonders of the last decade is the GPS satellite system. GPS stands for “Global Positioning System”, and if you buy a small GPS receiver (cost under $100), it will tell you your exact position on the Earth within a few meters. I’ve used such a receiver in the wilderness of Yosemite, in the souks of Fez, and in the deserts of Nevada. You can buy a car with a built-in GPS receiver that will automatically display a map on your dashboard showing precisely where you are. The military uses GPS to make its smart bombs land at just the location they want.
The GPS receiver picks up signals from several of the 24 orbiting GPS satellites. It is able to determine the distance to each satellite by measuring the time it took the signal to
(^7) At 1600 miles per hour, it will go 200 miles in 1/8 of an hour = 7.5 minutes.
go from the satellite to the receiver. Once it knows the distance to three satellites, it can then calculate precisely where on Earth it is.
The GPS satellites were not put in geosynchronous orbit, because the great distance would require that their radio transmitters have much more power to reach the Earth. They were not put in low orbit (LEO) because they would then often be hidden from your receiver by the horizon. So they were placed in a medium Earth orbit (MEO) about 12,000 miles high. They orbit the Earth every 12 hours.
To understand how GPS works, consider the following puzzle. A person is in a U.S. city. He is 800 miles from New York City, 900 miles from New Orleans, and 2,200 miles from San Francisco. What city is he in?
Look on a map. There is only one city that has those distances, and that is Chicago. Knowing three distances uniquely locates the position. GPS works in a similar manner, but instead of measuring distances to cities, it measures distances to satellites. And even though the satellites are moving, their locations when they broadcast their signals are known, so the computer in your GPS receiver can calculate its position.
Using gravity to search for oil
It was said earlier that every object exerts a small gravitational force on every other object. Remarkably, measurement of such small forces has important practical applications. If you are standing over an oil field, the gravity you feel will be slightly less than if over solid rock, for the simple reason that oil weighs less, and so its gravity isn’t as strong.^8 Such small gravity changes can even be measured from airplanes flying above the ground. An instrument can make a “gravity map” that shows the density of the material under the ground. Maps of the strength of gravity, taken by flying airplanes, are commonly used by businesses to search for oil and other natural resources.
A more surprising use of such gravity measurements was to make a map of the buried crater on the Yucatan peninsula, the crater left behind when an asteroid killed the dinosaurs. The crater was filled in by sedimentary rock that was lighter than the original rock, so even though it is filled, it shows a gravity “anomaly,” i.e. a difference from what you would get if the rock were uniform. An airplane flying back and forth over this region made sensitive measurements of the strength of gravity, and they produced the map shown below. In this map, the tall regions are regions in which the gravity was
(^8) The gravity of a spherical objects acts as if it is all coming from the center of the Earth.
That is true only if the object’s mass is uniformly distributed. If there is an oil field, then you can mathematically think of that as a sum of a uniform Earth, and a little bit of “negative” mass that cancels out some of the gravity. If you are close to the oil field, you will sense the reduced gravity because this little bit of negative mass will not attract you as much as if it were denser rock.
Measuring the Mass of the Earth
Have you wondered how we know the mass of the Earth? You can’t just put it on a scale. The first person to determine it was Henry Cavendish, in 1798. He did it in a very indirect way. Even though Newton had discovered the equation of gravity, at the time of Cavendish nobody knew the value of the constant G. Cavendish determined it by taking several masses in his laboratory and measuring the force of gravitational attraction between them. (It was not an easy experiment; as we said earlier, the force you get is about the same as the weight of a paramecium.) Once he knew G , he could figure out the mass of the Earth from the known distance to the moon, and the fact that it goes around once every 28 days. When asked what he was doing in his lab, his answer (some people claim) was “weighing the Earth!”
Problem: How could you use similar principles to determine the mass of the Sun?
Gravity on the Moon and Asteroids
Our Moon has about 1/81 of the mass of the Earth. So you might think its gravity would be 81 times less. But its radius is only 1/3.7 that of the Earth. Remember that gravity is an inverse square law, so from the small radius you would expect the force to be
^ 3.7
2 13.7 times larger. If you combine these two effects, you get the surface gravity is
81 ^
1 6 that on the Earth. And that is the value the astronauts found when they landed there. The gravity was so weak that they seemed to bounce around in slow motion. When they jumped, they went high, and when they came back down, they came down slowly.
What is the surface gravity of an asteroid, which has a radius of only 1 km? We can’t say, since we don’t know the mass. However, if you assume that the density is the same as for the Earth, then we can derive a simple result: the surface gravity is proportional to the radius.^9 Since the radius of the Earth is 6378 km, this means that the surface gravity on an asteroid is 1/6378 of the value on the Earth. If you weigh 150 lb on Earth, you would weigh 150 lb/6378 = 0.023 lb, or about a third of an ounce. That’s about the weight of three pennies on the Earth.
As we’ll show in a later section (on escape velocity), you have to be very careful, because your escape velocity would be very low. To jump into space from the Earth requires a velocity of 11 km/sec, but from the asteroid you would only require a velocity of 2 m/s.
(^9) That’s because the mass of the asteroid is
M density volume. Call the density
d. The volume of a sphere is
4
3 ^ R
- So the gravity at the surface is
F G
Mm R^2
Gm
M
R^2
Gm
4
3 ^ R
(^3) d
R^2
constants R^. So the weight of a mass of^ m^ = 1 kg on different planets will depend only on their radius R of that planet (if all planets are of similar material composition).
(The same jump speed would get you less than a foot high on the surface of the Earth.) You could easily launch yourself into space by jumping. This low escape velocity was a problem for a U.S. space probe called “NEAR” (stands for Near Earth Asteroid Rendezvous). If the satellite landed at a velocity of 2 m/s or more, then it might have bounced right back out into space.
Gravity in Science Fiction
One of the most common “errors” in Science Fiction movies is the implicit assumption that all planets in all solar systems have gravity about equal to that on the Earth. There is no reason why that should be so. Pick a random planet, and you are just as likely to be a factor of 6 lighter (and bouncing around like astronauts on the Moon) or six times heavier and unable to move because of your limited strength. Imagine a person who weighs 150 lb on the Earth, trying to move on a planet where he weighs 900 lb.
Numerical exercise : Use the simple radius formula to estimate the surface gravity on the moon. How close is your answer to the correct one that we calculated? Why aren’t the answers the same? Can you guess how far wrong the approximate formula must have been when applied to the asteroid?
Falling to Earth
Let’s now talk about everyday gravity, the gravity that you feel when you are near the surface of the Earth. When you jump off a diving board (or a bungee tower) gravity pulls you downwards. The force of gravity acts on all parts of your body, and makes them fall faster and faster. You accelerate, but in a very remarkable way: every second that you fall, you pick up an additional 9.8 meters per second of velocity. Put in the form of an equation, your velocity v after a time t is
v gt
where the constant g = 9.8 m/s^2 is usually called the acceleration of gravity.
By acceleration we mean the rate at which your velocity changes. For gravity, the acceleration is constant. After one second, your velocity is 9.8 meters/sec. After two seconds, it is 19.6 m/s. After three seconds, 29.4 m/s. After 4 seconds, 39.2 m/s. Every second you pick up an additional 9.8 m/s.
Here is a useful conversion factor: 1 m/s = 2.24 mph.
After falling for 4 seconds, our velocity was 39.2 m/s. To convert this to mph, just multiply by 2.24. So the velocity after 4 seconds is
39.2 ms 2.24 mphm / s 88miles per hour.
t
2 D
g
2 3000 m 9.8 ms 2
24.7 s
and its velocity would have been
v gt 9.8 ms 2 24.7 s 242.5 ms 543.2 mph
Ouch! That would be deadly. However, we made the mistake of neglecting the force of air as the packages were dropped. Every time the package hits a molecule of air, it transfers some of its energy. The force F of air depends on the area A of the package (more area means it hits more air molecules). That’s not surprising. But more interesting is the fact that the force depends on the square of your velocity. Double your velocity, and the force goes up four times; go 3 times faster, and your force increases by 9. This means that air resistance becomes extremely important at high velocities. This fact is the key to understanding not only dropped food, but also parachutes, space capsule reentry, and automobile gasoline efficiency.
The air resistance equation is:
F 12 A v^2
The symbol is the density of air, 1.25 kilogram per cubic meter, A is the front area of the object (as seen from the direction it is heading; if it is falling, then this is the area looking up at it), v is the velocity (in meters per second). F is the force of air resistance in Newtons. (See footnote 2 if you want to convert to pounds of force.)
As the object falls faster and faster, the force of air resistance gets greater and greater. Gravity is pulling down, but the air is pushing up. The force of the air resists the gravity; it opposes it. If the object keeps on accelerating, eventually the force of air will match the weight of the object. When that happens, gravity and air resistance are balanced. The object doesn’t stop moving, but it stops accelerating, that is, it no longer gains additional velocity. When this happens we say the falling object has reached terminal velocity.
For the food dropped over Afghanistan, the terminal velocity was about 9 mph. For a falling person it is typically 70-100 mph. That’s fast, but people have survived falls from great heights into water. (Try to imagine this next time you are going in a car or train at 74 mph.) If a falling person spreads out his arms and legs (like sky divers) that increases the effective area, and the person will fall even slower. A person using a parachute is not much heavier, but the parachute area A is large, so the terminal velocity is only about 15 mph. For King Kong (about the same area as a parachute, but much heavier) the terminal velocity was several hundred mph; he hit the ground without ever reaching it.
It is interesting to think about why it goes at this speed. If the falling object went any faster, the upward force (from the air) would be greater than the weight, and that net force would slow the fall. If the packet fell any slower than this terminal velocity, then the downward pull of gravity would be stronger than the upward force of air resistance, and the packet would fall faster.
The terminal velocity equation. If we take the air resistance equation and set
F mg (where m is the mass of the falling object) and solve for v, we get the equation for terminal velocity:
v 2 mg A
A food packet in Afghanistan had a mass of about m = 0.1 kg, and an area of about
A 0.3 m 0.3 m 0.1 m^2 ; g = 9.8 m/s^2. Putting in these numbers we get for the food packets:
v 2 0.1 kg 9.8 m / s^2 0.1 m^2 1.25 kg / m^3
4 ms 9 mph
That’s about the speed at which you jog. So the food floats down relatively slowly, not at the high speed we had obtained when we neglected air resistance.
Let’s calculate the terminal velocity for a person. We need to estimate the person’s area and his mass. The only area that counts is the area hit by air as he is falling. If he is diving, that would be the area of the top of his head. If he is doing a swan dive, it would be the area of the front of his body, roughly his height times his width. To make the numbers simple, let’s assume he is 2 meters tall and 1/2 meter wide, giving an area of
2 m 0.5 m 1 m^2. Take his weight to be M = 160 lb = 70 kg. Then, according to our formula, his terminal velocity will be:
v 2 mg A
2 70 kg 9.81 m / s^2 1 m^2 1.25 kg / m^3
33 ms 74 mph
With a parachute, the mass of the falling person is about the same (parachutes are light and don’t contribute much to the total weight), but the area subject to air resistance can be 30 times larger than the area of a person’s body. Plug that into the equation and see how much it slows the falling person. (Hint: it will be
30 5.5times slower. Do you see why?)
What about King Kong? Did he slow down? The answer is no – and the reason is interesting. If he were 10 times taller than a person, his weight would be 1000 times more. (He is not only 10 times taller, but also 10 times wider and 10 times thicker; that makes his volume 1000 times larger.) But his area would have been only 100 times greater. (The area that the air it hitting, to slow him down, is width times height; it doesn’t depend on his thickness.)
Figure: Aerodynamic design for a truck cab The top of the cab has had a contoured shape added to it to make the air bounce off smoothly, at an angle, instead of hitting the flat face of the truck head on. (The added shape is called “fairing.”) This is shape change is sometimes called “aerodynamic smoothing” and it saves gasoline. It makes the effective value of “ A ”, the area in the air resistance equation, smaller.
Note also that at half that speed (i.e. if v = 15 m/s = 33 mph) the force is four times less. That means that you use 4 times less gasoline to overcome air resistance. So you save even more gasoline by driving slower.
Force and Energy
The relationship between energy and force is remarkably simple: energy is force times distance. The equation is:
E F D
This means that if you push something with a force F (in Newton) for a distance D (in meters), then the energy it takes is E, joules. How much energy does it take to go up a flight of stairs? Suppose you go up 4 meters (about 13 feet) and you weigh 160 lb (or 714 N). Then the work you do is
E 714 N 4 m 3000 J 0.7 Cal
That’s why it’s hard to lose weight through exercise. Drink one 12 ounce can of cola, which typically has 140 calories, and you can work it off by going up 200 flights of stairs! Food contains a great deal of energy. That’s why you can get all the calories you need per day from about 2 lb of food. (1 kg of whole wheat bread contains about 2500 Cal.)
Force and Acceleration
If you push on an object, and there is no friction to hold it back, then it gains velocity. If you push twice as hard, it gains twice as much velocity. The acceleration is proportional to the force. But it also depends on the mass of the object. If the object has twice the mass, it needs twice the force to get it moving. These two facts are summarized the following equation:
Newton’s Second Law:
F ma
In this equation, m is the mass (in kg), a is the acceleration (in m/s^2 ). The force is then in Newtons^13.
This equation tells you how much things speed up – or slow down -- when you apply a force.
This is Newton’s Second Law. You may be wondering what Newton’s First Law is. It states that unless an object has an outside force on it, it will tend to keep its motion unchanged. But that is just a special case of the Second Law, since if F = 0, there is no acceleration, and that means no change in velocity. Nobody would ever teach the First Law these days if not for the fact that students sometimes wonder what came before the Second Law.
The g
In this book, accelerations are typically measured in m/s per second (m/s^2 ). For automobiles, we frequently measure acceleration in miles per hour every second, sometimes written as mph/sec. For example, if a car salesman tells you that a car will “go from zero to sixty in ten seconds” what he is really telling you is the acceleration: 60 mph in 10 seconds. That is the same as 6 mph every second = 6 mph/s.
Another very useful unit for acceleration, used in the military and by NASA, is the “ g ”, pronounced “gee.” One g is 9.8 meters/sec every second, i.e. it is the acceleration of gravity. Suppose an automobile accelerated at one g. How fast would it be going in 10 seconds? The answer is found from our velocity equation by setting
a 9.8m/s^2 :
(^13) Optional footnote: Some physics texts will take this equation as the definition of mass.
In that case, Newton’s Second Law can be stated as follows: “the mass of an accelerating object is approximately constant.” As we will see in the chapter on Relativity, at extremely high velocities, this law breaks down; mass increases with velocity.
they have a bigger force – their weight! So big objects will accelerate at the same rate as less massive objects.
The g-rule
There is a very good reason to think of acceleration in terms of g s: it enables you to solve important physics problems in your head. Suppose you are accelerated in the horizontal direction by 10 g s. How much force will that take? The answer is simple: 10 times your weight! When astronauts are accelerated by 3 g s, the force on them to do this must be 3 times their weight. We can call this the “ g - rule”: the force to accelerate an object is equal to the number of gs times the weight
To get the number of g s, just calculate the acceleration in m/s per second, and divide by 9.8. So, for example, we write that an acceleration of a = 19.6 m/s per second = 2 g. This is an acceleration of two g s. To accelerate something to 2 g s takes a force equal to twice the weight of that something.
Why the g-rule is true
The force required to accelerate an object of mass m is given by Newton’s Second Law:
F = m a
The number of gs is N = a/g so N g = a The weight of the object is w = m g We now use all of these equations: F = m a = m g N = w N
The last equation is the g-rule.
The distance equation
In the beginning of this chapter, we said that the distance you fall in a time t , under the influence of gravity, is
D 12 gt^2
The same formula will apply when you undergo any constant acceleration a , if you simply replace the symbol g with the symbol a. So the distance you travel will be:
D 12 at^2
We can write this in terms of the final velocity v by substituting t = v/a :
D 12 at^2
12 a v a
2
12^ v^
2 a
If we solve this equation for a , we get
a v^2 2 D
This equation tells you the acceleration you need to reach a velocity a in a distance D.
Shooting an astronaut into space: the rail gun
To go into orbit around the Earth, a satellite must have a velocity of 8 km/sec. Why not give it this velocity in a gun? Could we literally “shoot” an astronaut into space?
The answer is: you might be able to do this, but the astronaut would be killed by the force required to accelerate him. Let’s assume we have a very long gun, an entire kilometer long. If we use the equation in the last section, we derive that the required acceleration is a = 3200 g , i.e. the acceleration is 3200 times the acceleration of gravity.
Calculation: 1 km gun shooting to 8 km/sec. We can calculate this by using the last formula in the last section. The hardest part is getting the units right. The distance D = 1 km = 1000 meters, and the velocity v = 8 km/sec = 8000 m/sec. Plugging these in, we get:
a v^2 2 D
(8000 ms )^2 2 1000 m 32000 ms 2
(^32000) sm 2 9.8 (^) sm 2 ^ g 3200 g
