6 Coursework Power & Exponential Relationships Breithaupt pages 247 to 252 September 11 th, 2010...
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Transcript of 6 Coursework Power & Exponential Relationships Breithaupt pages 247 to 252 September 11 th, 2010...
The equation of a straight line
For any straight line:y = mx + c
where: m = gradient = (yP – yR) / (xR – xQ)
and c = y-intercept
The power law relationshipThis has the general form:
y = k x n
where k and n are constants.
An example is the distance, s travelled after time, t when an object is undergoing acceleration, a.
s = ½ at 2
s = y; t = x; 2 = n; ½ a = k
To prove this relationship:– Draw a graph of y against x n – The graph should be a straight line
through the origin and have a gradient equal to k
y
x n
gradient = k
Common examplespower, n = 1: direct proportion relationship: y = k x – prove by plotting y against xpower, n = 2: square relationship: y = k x2 – plot y against x2
power, n = 3: cube relationship: y = k x3 – plot y against x3
power, n = ½: square root relationship: y = k x ½ = k √x – plot y against x ½ power, n = - 1: inverse proportion relationship: y = k x -1 = k / x – plot y against 1 / x power, n = - 2: inverse square relationship: y = k x -2 = k / x2 – plot y against 1 / x2
In all these cases the graphs should be straight lines through the origin having gradients equal to k.
QuestionQuantity P is thought to be related to quantities Q, R and T by the following equation: P = 2π Q R 2
T 3
What graphs should be plotted to confirm the relationships between P and the other quantities?
State in each case the value of the gradient.
When n is unknownEITHER - Trial and error Find out what graph yields a straight line. This could take a long time!
OR - Plot a log (y) against log (x) graph.Gradient = ny-intercept = log (k)
Logarithms
Consider:
10 = 10 1 100 = 10 2 1000 = 10 3
5 = 10 0.699 50 = 10 1.699 500 = 10 2.699
2 = 10 0.301 20 = 10 1.301200 = 10 2.301
In all cases above the power of 10 is said to be the LOGARITHM of the left hand number to the BASE OF 10
For example: log10(100) = 2 log10(50) = 1.699 etc..(on a calculator use the ‘lglg’ button)
Natural Logarithms
Logarithms can have any base number but in practice the only other number used is 2.718281…,
Napier’s constant ‘e’.
Examples: loge(100) = 4.605 loge(50) = 3.912 etc..
(on a calculator use the ‘ln’ button)
These are called ‘natural logarithms’
Multiplication with logarithms
log (A x B) = log (A) + log (B)
Example consider: 20 x 50 = 1000
this can be written in terms of powers of 10:
10 1.301 x 10 1.699 = 10 3
Note how the powers (the logs to the base 10) relate to each other:
1.301 + 1.699 = 3.000
Division with logarithms
log (A ÷ B) = log (A) - log (B)
Consider: 100 ÷ 20 = 5
this can be written in terms of powers of 10:
10 2 ÷ 10 1.301 = 10 0.699
Note how the powers relate to each other:
2 - 1.301 = 0.699
Powers with logarithms
log (An) = n log (A)
Consider: 2 3 = 2 x 2 x 2
this can be written in terms of logs to base 10:
log10 (2 3) = log10 (2) + log10 (2) + log10 (2)
log10 (2 3) = 3 x log10 (2)
Another logarithm relationship
log B(Bn) = n
Example: log10 (10 3) = log10 (1000) = 3
The most important example of this is:
ln (en) = n
[ loge (en) = n ]
How log-log graphs work
The power relationship has the general form:
y = k x n
where k and n are constants.
Taking logs on both sides:
log (y) = log (k x n)
log (y) = log (k) + log (x n)log (y) = log (k) + n log (x) which is the same as:
log (y) = n log (x) + log (k)
log (y) = n log (x) + log (k) This has the form of the equation of a straight line: y = m x + cwhere:y = log (y) x = log (x) m = the gradient
= the power nc = the y-intercept
= log (k)
QuestionDependent variable P was measured for various values of independent variable Q. They are suspected to be related through a power law equation: P = k Q n where k and n are constants. Use the measurements below to plot a log-log graph and from this graph find the values of k and n.
Q 1.0 2.0 3.0 4.0 5.0 6.0
P 2.00 16.0 54.0 128 250 432
log 10 (Q)
log 10 (P)
Exponential decayThis is how decay occurs in nature. Examples include radioactive decay and the loss of electric charge on a capacitor.
The graph opposite shows how the mass of a radioactive isotope falls over time.
Exponential decay over time has the general form:
x = xo e - λ t
where:
t is the time from some initial starting point
x is the value of the decaying variable at time t
xo is the initial value of x when t = 0
e is Napier’s constant 2.718…
λ is called the decay constant.– It is equal to the fraction of x that decays in a unit time. – The higher this constant the faster the decay proceeds.
In the radioisotope example: t = the time in minutes.
x = the mass in grams of the isotope remaining at this time
xo = 100 grams (the starting mass)
e = Napier’s constant 2.718…λ = the decay constant is equal to the fraction of the isotope that decays over each unit time period (1 minute in this case). About 0.11 min-1 in this example.
Proving exponential decay graphically
x = xo e - λ t
To prove this plot a graph of ln (x) against t .
If true the graph will be a straight line and have a negative gradient.Gradient = - λ
y-intercept = ln (xo)
NOTE: ONLY LOGARITMS TO THE BASE e CAN BE USED.
How ln-t graphs workExponential decay has the general form:
x = xo e - λ t
Taking logs TO THE BASE e on both sides:
ln (x) = ln (xo e - λ t)
ln (x) = ln (xo ) + ln (e - λ t)
ln (x) = ln (xo ) - λ twhich is the same as:
ln (x) = - λ t + ln (xo )
ln (x) = - λ t + ln (xo ) This has the form of the equation of a straight line:y = m x + cwith:y = ln (x) x = t
m, the gradient = the negative of the decay constant = - λ
c, the y-intercept = ln (xo )
QuestionThe marks M of a student are suspected to decay exponentially with time t.They are suspected to be related through the equation: M = Mo e – k t. Use the data below to plot a graph of ln(M) against t and so verify the above statement. Also determine the student’s initial mark Mo (t = 0 weeks) and the decay constant k, of the marks.
t / weeks 1 2 3 4 5 6
M 72 59 48 40 32 27
ln (M)