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## Numb3rs 109: Sniper Zero | ||||||||

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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
v_{vertical}(t). | ||||||||

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- Recall that the acceleration of an object is equal to the instantaneous change in velocity, i.e. a(t)=v'(t). Apply the fundamental theorem of calculus to this equality to deduce that , where g is the acceleration due to gravity.
- We can do even better by applying the same trick to velocity. Namely, we know that v(t)=y'(t), where y(t) is the vertical position of the object at time t. Apply the fundamental theorem of calculus again to show that
- Now write down an equation for the
*horizontal*position x(t) of the bullet in terms of the initial horizontal velocity v_{horizontal}(0). (Hint: remember, we disregard drag and wind so the only acting force is gravity.)
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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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- What is the maximum height achieved by the bullet? At what time is this height achieved? (Hint: what is the vertical velocity of the bullet when it's at a peak height?)
- What is the horizontal distance of the bullet from the origin at the time of peak height? What is the distance to the point at which the bullet is again at the height from which it was fired, that is, y=0?
- Show that the trajectory of the bullet is a parabola.
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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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< < | equation:, where k is a constant and dx/dt is | |||||||

> > | 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 | ||||||||

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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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- Suppose yesterday you heard that annual inflation was 3% in the last year. If x is the price of a representative basket of goods, and t is measured in years, what is the corresponding proportionality constant k in the exponential growth equation that models the price increase? (Hint: note that in this case dt=1 year.)
- What if t is measured in days instead?
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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=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 | |||||||

> > | 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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< < | Activity 4: | |||||||

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## Activity ##a#.: | |||||||

- Find a constant r so that 2
^{t}=e^{rt}. - Show that 2
^{t}becomes larger than any polynomial in t, for sufficiently large t. (Hint: suppose p(t)=t^{n}for some positive integer n. For which t is 2^{t}> p(t)?) - Can you think of a function f(t) which grows faster than an exponential function, in the sense of part 2 above?
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In practice, when talking about compound interest two quantities are | ||||||||

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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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- You open a savings account which earns 2% interest with a deposit of $1000. Would you rather the interest compound daily or monthly? Write down the formula for the amount of money in the account after a year in both cases. (Hint: write down the expression for the amount of money after one period of compounding, now after two periods (don't simplify!), then three... See the pattern?)
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< < | - Suppose we decide to compound not once a month or a day, but once every split second. In fact, we can let the number of compounding periods go to infinity, thus letting the length of each period approach zero. Use the fact that 0.02*t dollars. This is continuous compounding.
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> > | - Suppose we decide to compound not once a month or a day, but once every split second. In fact, we can let the number of compounding periods go to infinity, thus letting the length of each period approach zero. Use the fact that 0.02*t dollars. This is continuous compounding.
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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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