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Numb3rs 109: Sniper Zero
Ballistic Trajectory
A 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.  
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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 reaching the starting elevation. The force of gravity acts on the bullet, creating a downward acceleration of g=9.8 m/sec^{2} and so influencing the vertical component of the velocity vector (see diagram below) over time. Since we disregard drag, the horizontal component of the velocity does not change.  
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The next activity involves figuring out the equations describing the speed and position of an object in freefall. These derivations make some use of calculus. Try to follow them and do the exercises, but if you can't, just use the equations mentioned in order to do activity 2.  
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Activity ##a#.:Let us first consider only the vertical direction of motion. For the sake of brevity, I'll just write v(t) below instead of v_{vertical}(t).
Activity ##a#.: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. Furthermore, mathematically recreating the path of a bullet after it has hit a target, thus only knowing its angle of entry, is much harder.
Exponential Growth
Here 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. ( Wellknown 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 longterm debt in the form of lavish salary increases and exponential growth in new, highly compensated faculty and staff directly  
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> >  reporting to him." ( UCSF dean is fired, cites whistleblowing, Los Angeles Times, Dec 15,  
2007)
While the above excerpts describe growth in entirely different areas,  
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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  
> >  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:, where k is a constant and dx/dt is
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
often exponential growth occurs in situations where "x creates more x",
typical examples being population growth and compound interest.
Exponential growth can occur both when the time intervals are
discrete, for example in annual or monthly interest compounding, and
when the time variable is continuous, as in continuous
compounding of interest or modelling of large
populations. The discrete time case is more often encountered in
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 time variable t is continuous, we can solve the differential equation . By separating variables we get , integrating we arrive at , where C is some constant, and exponentiating both sides, we finally get x=De^{kt}, where D is a constant. We can solve for D by plugging in t=0, the starting  
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< <  time, to arrive at the general solution . Exponential growth is much faster than polynomial, as the example below illustrates in case of e^{t} versus t^{3}.  
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Figure ##f#.:  
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> >  Exponential growth is much faster than polynomial, as the figure illustrates in case of e^{t} versus t^{3}.  
Activity ##a#.:
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 ##a#.:
 
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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 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 (remember that strange function used in logistical regression?). While similar in the low range to the exponential function, sigmoidal growth reflects the fact that at some point growth must slow down due to lack of resources.  
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