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figure out which is being meant from the context: just ask yourself, is the sentence talking about a vector or a scalar? | ||||||||
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Let us first consider only the vertical direction of motion. For the sake of brevity, I'll just write v(t) below instead of vvertical(t). | ||||||||
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Suppose the bullet is fired at an angle of 30 deg as in the first picture, with a speed of 900m/s from an initial point x(0)=0 and y(0)=0 (i.e. from the origin) at time t=0. Use the | ||||||||
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Analysing the general situation, in which both wind and drag affect | ||||||||
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Here are a few recent uses of the term exponential growth in the news media: | ||||||||
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The company has had a spectacular two years, riding the exponential growth in oil prices that helped to increase profits | ||||||||
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( Business Big Shot: Alasdair Locke , The Times, Dec 20, 2007) | ||||||||
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After years of exponential growth, there has recently been a slow down in the Northern Ireland property market. ( Well-known property firms merge , BBC News, Dec 7, 2007) | ||||||||
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Kessler himself came under university scrutiny for alleged financial irregularities. In January 2005, an anonymous source contended he "spent or formally committed all of the reserves of | ||||||||
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the one thing they have in common is the use of the term _exponential growth._ In mathematics, we say that quantity x grows exponentially with respect to time t if x satisfies the following differential | ||||||||
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either a derivative, when t is continuous, or the change in x in a given time interval, when t is discrete. In plain words, this means that x grows exponentially if it increases proportionally to its own value. Most | ||||||||
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practice and is easier to analyse mathematically, since we don't need to resort to the exponential function. | ||||||||
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The reason why such growth is called exponential is that when the | ||||||||
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where C is some constant, and exponentiating both sides, we finally get x=Dekt, where D is a constant. We can solve for D by plugging in t=0, the starting | ||||||||
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faster than polynomial, as the example below illustrates in case of
et versus t3.
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year, with such monthly compounding, you'll owe $121.94. Might not seem like a huge difference from the once a year compounding sum of $120, but over longer periods of time, the difference becomes substantial. | ||||||||
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In popular usage, the expression "exponential growth" is often used as a synonym for "very fast growth". There's no good reason to describe faculty hiring practices, as the third quote |
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approximate the actual trajectory is to ignore the effects of drag and wind, instead looking only at gravity. | ||||||||
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Consider the figure to the right. A bullet leaves the barrel of a gun inclined at a 30 deg angle and flies a horizontal distance of d before | ||||||||
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The next activity involves figuring out the equations describing the
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direction of motion with magnitude equal to the speed of the object. However, in everyday usage and even in many physics textbooks the term velocity is used to denote both the vector and its magnitude, the speed. It is usually easy to figure out which is being meant from the context: just ask yourself, is the sentence talking about a vector or a scalar? | ||||||||
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Activity 1:
Let us first consider only the vertical direction of motion. For the
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Numb3rs 109: Sniper Zero
Ballistic TrajectoryA bullet, like any other object flying through the air, is subject to the forces of gravity, air resistance, and wind. One way to closely approximate the actual trajectory is to ignore the effects of drag and wind, instead looking only at gravity.![]() ![]() ![]()
Technically, the term velocity means the vector pointing in the
direction of motion with magnitude equal to the speed
of the object. However, in everyday usage and even in many
physics textbooks the term velocity is used to denote both the
vector and its magnitude, the speed. It is usually easy to
figure out which is being meant from the context: just ask
yourself, is the sentence talking about a vector or a scalar?
Activity 1:
Let us first consider only the vertical direction of motion. For the
sake of brevity, I'll just write v(t) below instead of
vvertical(t).
Activity 2: Suppose the bullet is fired at an angle of 30 deg as in the first picture, with a speed of 900m/s from an initial point x(0)=0 and y(0)=0 (i.e. from the origin) at time t=0. Use the equations from activity 1 to answer the following questions.
Analysing the general situation, in which both wind and drag affect
the path of a bullet, is in fact very complicated. You can get a taste
of the difficulties involved by reading the wikipedia article on
external ballistics
![]() Exponential GrowthHere are a few recent uses of the term exponential growth in the news media:The company has had a spectacular two years, riding the exponential growth in oil prices that helped to increase profits by a fifth in 2006 to £28.5 million. ( Business Big Shot: Alasdair Locke , The Times, Dec 20, 2007) After years of exponential growth, there has recently been a slow down in the Northern Ireland property market. ( Well-known property firms merge , BBC News, Dec 7, 2007) Kessler himself came under university scrutiny for alleged financial irregularities. In January 2005, an anonymous source contended he "spent or formally committed all of the reserves of the dean's office and has also incurred substantial long-term debt in the form of lavish salary increases and exponential growth in new, highly compensated faculty and staff directly reporting to him." ( _UCSF dean is fired, cites whistle-blowing_, Los Angeles Times, Dec 15, 2007)While the above excerpts describe growth in entirely different areas, the one thing they have in common is the use of the term _exponential growth._ In mathematics, we say that quantity x grows exponentially with respect to time t if x satisfies the following differential equation: ![]() Activity 3:
The reason why such growth is called exponential is that when the
time variable t is continuous, we can
solve the differential equation
![]() ![]() ![]() ![]() ![]() Activity 4:
In practice, when talking about compound interest two quantities are
important. One is the annual interest rate, sometimes called the annual
percentage rate (APR). The other is the number of compounding periods per
year: how many times per year is the interest added to the principal
amount. For instance, say you have $100 credit card debt with an APR of
20%. Usually credit cards compound monthly, so there are 12 compounding
periods per year. Thus if you make no payments (and incur no additional
penalties or expenses) for a whole year, your debt will not simply
be 100+100*0.2=120, which it would if the interest was compounded
only once per year. Instead, after the first month, you'll owe
100+100*(0.2/12)=101.67 dollars. After the second month, you'll owe
101.67+101.67*(0.2/12)=103.36 dollars, and so on. At the end of the
year, with such monthly compounding, you'll owe $121.94. Might not seem
like a huge difference from the once a year compounding sum of $120, but
over longer periods of time, the difference becomes substantial.
Activity 5:
In popular usage, the expression "exponential growth" is often used as a synonym for "very fast growth". There's no good reason to describe faculty hiring practices, as the third quote in the beginning of this section does, in terms of exponential growth. While an exceptional number of faculty might have been added during Kessler's tenure as dean, there's no sense in which "faculty makes more faculty" proportionally to existing numbers. At other times, "exponential growth" can be more accurately described as sigmoidal
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