ASSIGNMENT 1
(To be done after studying Blocks 1 and 2.)
Course Code: MTE-14
Assignment Code: MTE-14/TMA-1/2006
Maximum Marks:100
1)
a)
Through one example each from the socio-economic environment bring out the difference
between
i)
ii)
iii)
iv)
b)
Linear and non-linear models
Static and dynamic models
Discrete and continuous models
Deterministic and stochastic models.
(4)
Give one example each of a mathematical model which is
i) Linear and Static
ii) Dynamic and stochastic.
Also, give justification for your answer
c)
(2)
Classify the following into liner and non-linear models, justifying your classification.
i) Simple harmonic motion for small amplitude of oscillation.
ii) Population growth model given by
dN
= aN (BkN), a, B, k are constants.
dt
iii) Equation for velocity v of a particle at any time t, moving with a constant acceleration
a, and initial velocity u.
iv) Equation describing dynamic stability of market equilibrium price given by
pt = [1 + k (apt1 +k(b – B)],a,b,A,B,k are constants

2)
3)

and pt is the price in period t.
(4)
a)
Give a recent situation/incidence from the real world where you would like to formulate
mathematical model to study the problem involved. State the type of modelling you will use
for this problem giving reasons in support of your answer. Further, list at least 3 essentials and
3 non-essentials for this problem.
(5)
b)
A particle is projected from the base of an inclined plane making an angle α with the
horizontal. The initial velocity of project is making an angle β with the horizontal.
i) Derive the equation to the trajectory.
ii) Find the point at which the particle strikes the plane.
(5)
a)
Deduce the dimension of G from Newton’s law of gravitation.
(2)
b)
A particle moves from an initial velocity u at a constant acceleration f. Find an expression for
the distance covered in time t using dimensional analysis.
(5)
c)
A particle of mass m moves on a straight line towards the centre of attraction starting from
rest at a distance a from the centre. Its velocity at a distance x from the centre varies as
a3  x3
x3
. Find the law of force.
(3)
3
4)
a)
Calculate terminal velocities of raindrops moving in motionless air with the following
diameters D=0.001, 0.005, 0.15, 0.2, 0.25. Use these calculations to make a graph of V term
plotted against diameter. What kind of curves do these points lie on?
(4)
b)
A particle of mass m is falling under the influence of gravity through a medium whose
resistance equals μ times the velocity. If the particle be released from rest, show that distance
fallen through time t is
 t

gm2   m
t 
1
e

m
2 


(6)
5)
a)


A particle is moving under a central attracting force F. Show that its equations of motion can
be written as
d 2u
F
u


2
2 2
d
h u


where u 
b)
1
d
.
and h  r 2
r
dt
(4)
Use this to write down the equation of motion of a particle under a force F=cu3, where c is a
constant. Solve the equation of motion. You will need to discuss three cases
c  h 2 , c  h 2 and c  h 2 . Discuss, in which of these cases, is a closed orbit possible.
6)
a)
A particle describes a central orbit rn = an cosn θ under a force directed towards the pole.
Find the law of Force.
1
(Rewrite the equation in terms of u  , take logarithms and differentiate with respect to θ.
r
2
d u

Calculate 
(3)
 u  and use 5a) to find F).
2
 d



b)
Consider a particle moving in a cardioid given by the equation r  a2 (1  cos ). Use part a)
to find the law of force.
7)
8)
(6)
(2)
c)
If a straight tunnel connects any two places on the surface of the earth and if a particle is
dropped into the tunnel from one place, show that it reaches the other place in 42 minutes
nearly, the radius of the earth being 6400 km. It is known that the acceleration of a body at any
point inside the earth varies as its distance from the centre of the earth.
(5)
a)
Find a relationship between the escape velocity and the minimum velocity with which a
particle is to be projected horizontally so that the particle circles around the earth.
(4)
b)
Explain the term ‘plume rise’ in the context of industrial pollution caused by effluents arising
out of a single stack. How is it different from the “diffusion” of plume.
(4)
c)
Can we use the Gaussian model equation which describes the motion of air pollutants for
studying vehicular pollution also? Give reasons for your answer.
(2)
a)
Sulphur dioxide is emitted at a rate of 150 g/s from a stack with an effective height of 50 m.
The wind speed at a stack height is 5 m/s and the atmospheric stability class is C as in Unit 6,
Block 2, for the day. Determine the ground level concentration along the centre line at a
4
b)
9)
a)
distance (I) 400 m (II) 2000 m from a stack, in micrograms per cubic metre.
(6)
For the non-linear model of motion of simple pendulum find the period of the pendulum if the
pendulum is started at rest from an arbitrary angle 40o given that the length of the pendulum is
30 cm.
(4)
A particle is moving under a central force along the path given by the equation r  a tan .
Show that the radial acceleration is
10)
k 2 
2a 2 
d
where, k  r 2
.
3
3 
2 
dt
r 
r 
(5)
b)
For the problem of finding the period of oscillation of a simple pendulum, obtain a
formulation in the case when air resistance is proportional to the square of the velocity. Find a
solution for this formulation.
(5)
a)
Consider the motion of a simple pendulum where a body
O
of mass m is attached to a string at the point O (see Fig.1).
The force acting on the body are (i) the weight mg of the
θ
body acting vertically downwards at A ii) the tension
T
T=μx along the string from A towards O.
dx
iii) resistance force R  f
,
A
dt
proportional to velocity iv), external force F=L sin pt.
Formulate the equation of motion where both f
x
mg
and μ are positive constants. Also solve this equation of motion.
Fig.1
b)
(6)
Write down the equation, that models the growth of phytoplankton under the following
assumptions:
i)
Phytoplankton be limited to grow in two-dimensional region, say along y and z-axis.
ii) wind velocities are in y and z-directions only and the wind speeds are constant in all
directions.
iii) motion is in steady state.
iv) diffusivity is constant.
Explain each variable of the equation in the context of the problem.
5
(4)
ASSIGNMENT 2
(To be done after studying Blocks 3 and 4)
Course Code: MTE-14
Assignment Code: MTE-14/TMA-2/2006
Maximum Marks: 100
1)
a)
Consider the population x(t) of fish and y(t) of sharks at any time t in a certain region of the
pacific ocean. Sharks prey on fishes and their population would decrease if isolated from the
fish population. Let us assume further the following assumptions.
1) The change in the shark and fish populations, in isolation, is respectively proportional to the
present population of sharks and fish.
2) The number of sharks and fish caught by fishermen is directly proportional to the present
population of the shark and fish population respectively. The fishing methods do not discriminate between sharks or fish.
3) The number of fish eaten by sharks is directly proportional to the product of the number of fish
present and the number of sharks present.
4) The additional number of sharks surviving is directly proportional to the number of fish eaten.
Under assumptions1) – 4)
i)
Formulate the mathematical model for the given problem and write a system of
differential equations governing it.
ii)
Is the system of equations obtained in i) an autonomous system? Give reasons.
iii)
Find the steady-state solution of the system in i) above. What does the solutions obtained
correspond to?
iv)
From the system of equations obtained in i) obtain the expression for dy/dx and hence
solve it under the condition x(0) = x0, y(0) = y0.
(12)
b) Consider a closed population of homogeneously mixing individuals with no removals. Suppose
that a(a > 0) number of infectives are introduced to a group of n susceptibles at time t=0 and
infection spreads by the contact between the infectives and the susceptibles.
2)
i)
Formulate the mathematical model for the given problem and write a differential equation
governing it.
ii)
Find the number of susceptibles and infectives at any time t.
iii)
Find the time when the rate of appearance of new infectives is maximum and also the
density of susceptibles at that time.
(8)
a) In a perfectly competitive market, the supply function S(p) and demand function D(p) are given
as follows:
p 2  17
; D  p    p  5, where p is the price:
4
Find the equilibrium prices
S  p 
i)
ii)
Using Walrasian stability condition, determine whether the equilibrium prices are stable.
(4)
6
b) Find the current density when oxygen diffuses through a red cell of thickness 2.5 x 10-8 cm, the
two ends of which are maintain at a fixed concentration C0.
(3)
c) Let the slab represents a biological cell in a large bathing solution of solute with fixed (given)
concentration of 13 mg. Then find the concentration distribution inside the cell at any given
position 0 < x < h and time t > 0.
(3)
3)
Consider the group of individuals born in a given year (t = 0) and let n(t) be the number of these
individuals surviving t year later. Let x(t) be the number of members of this group who have not
had smallpox by year t and are therefore still susceptible. Let β be the rate at which susceptibles
contract smallpox and let ν be the rate at which people who contract smallpox die from the disease.
Finally, let μ(t) be the death rate from all causes other than smallpox. If dx/dt and dn/dt are
respectively the rates at which the number of susceptibles and entire population decline due to
contraction from smallpox and also due to death from all causes then
i)
Formulate the above problem by writing equations for dx/dt and dn/dt.
ii) Taking z = x / n, show that z satisfies the initial value problem
dz
  z (1  z), z(0) 1
dt
iii) Find z(t) at any time t.
1
. Using these values, determine the proportion of 20 years
8
old who have not had smallpox.
(10)
iv) Bernoulli estimated that ν = β =
4)
a) The following table gives the sales of an item produced by a company since 1999.
Year
1999
2000
2001
2002
2003
2004
Sales (in crores of rupees)
10
12
9
13
12
14
Obtain the least square trend-line equation by using 1999 as zero year.
(5)
b) Let a, K and N0 be positive real numbers such that 0 < N0 < K. Find the solution of the logistic
equation
dN
N 

 aN 1 
 , N (0)  N0
dt
K 

If N0, N1 and N2 be the values of N at t = 0, t = T and t = 2T respectively, then show that
N1 aT
K  N1
e

N0
K  N0
N 2 2aT K  N 2
e

N0
K  N0
Also if at time t = t1, the population N reaches half its carrying capacity then what is the form of
the solution of the logistic equation?
(5)
7
5)
A monopolist sets a price ‘p’ per unit and the quantity demanded ‘q’ is given by the following
relation:
q = 17 – p.
Let there be a fixed cost of Rs.9 and a marginal cost of Re.1 per unit.
i)
Write the profit function of monopolist.
ii) For maximum profit, find the number ‘x’ of units produced. Also find the maximum profit.
iii) A potential entrant enters into the business of the monopolist. He believes that the monopolist
will go on making ‘x’ units. Write the profit function of the entrant.
iv) For maximum profit of entrant, find the number ‘z’ of units produced.
v) Find the maximum profit of the entrant. Explain whether he should enter into the business or
not.
vi) Find the profit made by the monopolist after the entrant has entered into business.
6)
(10)
Consider the system of differential equations
dx
dy
 y,
 x  2 x3
dt
dt
i)
Is the system autonomous? Give reasons.
ii) Find the nature of critical points of the system.
iii) Sketch the trajectory for the corresponding linear system. Show that the only trajectory on which
x  0, y  0 as t   is y  x and the trajectory for which x   , y   as t   is y = x.
iv) Determine the trajectory for the non-linear system.
7)
a)
(10)
Solve the Gompetz equation
dN
K
 rN ln   , N(0)  N0
dt
 N
i)
For the data r = 0.71 per year, K = 80.5  106 kg.,
N0
 0.25 , use the model to find the
K
predicted value of N(2).
ii)
For the same data use the model to find time  at which N () = 0.75K.
(5)
b) Consider the flow of fluid through a rigid tube of circular cross-section for
which Poiseuille law is applicable. The radius of the tube is given to be 8  10–3
cm., pressure drop P is 6  103 dynes/cm2, viscosity μ = 0.027 poise. In a
laboratory experiment it has been found that the maximum velocity of flow is
1.185 cm/sec. then find I) the length of the tube ii) the rate of flow iii) resistance to
the flow.
(5)
8)
a) The short-run cost function for an entrepreneur is q3 –7q2 + 16q + 90.
Determine the price at which the entrepreneur ceases production in an ideal market.
Derive also the supply function.
(5)
8
b) Solve the following two person zero-sum game by using dominance
Player B
Player A
9)
I
II
III
IV
I
3
5
8
4
II
5
6
7
2
III
4
3
9
8
IV
10
7
9
5
V
6
9
8
3
a) For a given set of securities, all their portfolios lie on or within the boundary of the
region shown in Fig.2.
Fig.2
In the feasible region, find a portfolio which has maximum return. Also, find a portfolio
in this region which has minimum risk.
(3)
b) Explain the method of delineating the efficient frontier of a feasible region.
(5)
c) Given all the portfolios of n securities what criterion would an investor use to select a
good portfolio?
(2)
9