QUESTION
As shown in the above curve the Phi being the y axis and time being the x-axis, the curve is exponential.
2.The curve which best plots the graph is . which has gradient equal to and the y intercept equal to 1.
3. a) We can use the logarithm principle to convert the equation:
SOLUTION
Solving, this equation for a and K we have a=16.44,k=0.067
So, Note: we used the first two coordinate values to get the results.
3.b) the population after 20 hours is 18.696, which is different from the table as because we choose two initial values for the calculation .
4. a) The growth rate of the bacteria population is
So, the growth rate of bacteria population after 20 hrsis 1.25 millions per hour, according to our model.
b) Actually the calculation in the above part has been done according to the model , which can be different from the actual experimental result. The better values can be calculated by using recursive methods of extrapolation of the curve drawn in the question 1.
c) The mathematical model is just an approximation of the experimental model. The better results of the calculation can be done through the numerical methods.
4. a) We can obtain the time when the population is minimal by the maxima and minima calculation as:
Which, is as
The result from the mathematical model yields=9.6 hr
b) The total population at this time is 39 million approx
5. Given equation:
So, taking log by base 10 on both the sides:
(since log e=0.43)
So, we have two simultaneous equations:
log(100)=logC+ d(2)*(0.43)
log(8)=logC+ d(9)*(0.43)
which, give C=206 and d=-0.361.
7.
We chose a plate of weight around 7.87 g. The terminal velocity of plate was found to be 3 m/s, before touching the ground.
8.) The value of the drag force constant k=
9.)
As shown in the above graph the velocity is represented by
10.) If we take the other coefficients of time equal to zero, the velocity becomes a constant which can not fit the mathematical model. Again if the coefficients of time other than A and B are zero then the velocity is a linear function of time, then the experimental values would be matching our results. The coefficient C if not equal to zero would make our velocity curve parabolic which matches the curve of a projectile motion. Hence, the realistic curve is equivalent to our model till velocity is represented as power of 2.
11.) The strength of the method is that it gives the exact curve for the velocity when the crunched paper is allowed to free-fall from the rest condition. Another, strength of the velocity-time relation is that it matches our mathematical model of drag force produced by air in the opposite direction of the velocity of the paper. The limitation can be counted that as that the curve shows dependency on the powers of time greater than 2.
12.) The crunched paper takes almost 0.33sec to attain the velocity of 2m/s.
13.) The velocity is given as:
So, differentiating the relation we get the acceleration as:
14.) The newton’s equation gives:
Here, Vt is the terminal velocity, U0 is the initial velocity, while a is the acceleration(which in this case is due to gravity ).
15.) From the given function of
So, we get acceleration
16.) The velocity in the earlier case was given as thepowers of time, where as in this case the velocity is exponential curve, with the power unit power of time.
KI01
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