A parachutist, whose weight (actually mass) is , drops from a helicopter
above the ground. She falls towards the earth under the influence of gravity. Assume that the gravitational force is constant. Assume that the force due to air resistance is proportional to the velocity of the parachutist. The proportionality constant is
when the parachute is closed, and is
when it is open. If the parachute does not open until
after the parachutist leaves the helicopter, after how many seconds will she hit the ground?
b) A projectile is fixed with a constant speed v at two different angles of projection and
such that it gives the same range. Show that
Let downward direction be taken as positive. The mass of the parachutist is (m=64) kg. The gravitational force is (mg=64\times 9.8=627.2) N. The resistive force is proportional to velocity and acts upward, so its magnitude is (kv). While the parachute is closed ((k_1=16) kg/s), the equation of motion is [ m\frac{dv}{dt}=mg-k_1 v. ] This is ______ ______ ____ _______ ________ __________ _________.
_______ ______ _________ ______ __________.
________ ________ _____ _____ _______ _________ ____ ____ _____ __________ ______.
__________ _______ ___ ______ _________ ___ _______ __________.
__________ _________ __________ __________ ______ ___.
_______ _________ ___ ________ ____ ______ _________ _______ _____ _________ ______ _______.
__________ ________ ____ ________ _______.
___ __________ __________ ___ ________ ________ _________ _______ ___ ________ __________ __________.
______ ______ __________ ___ __________ __________.
_______ _____ _______ ____ __________ _______ ________ _______ ______ _________ _____.
_______ _________ ____ _________ _____ __________ _______.
______ ___ __________ _________ ___ ___ ____ ________ _______ _________ ___ _______.
______ ______ ___ ______ ________ ______ ________ _______.
__________ ________ ___ ____ _____ _____ ______ ____ ______ _________ _______ ______.
______ ________ ______ __________ _____.
__________ ___ __________ ______ ________ ____ ______ ____.
________ _________ _________ __________ ________.
_________ _________ _____ _____ _________ _______.
_____ ______ ______ _________ _________ ________ ____ _____ _____ _____ __________.
_______ ___ ___ ______ _____ ____ ____ _______.
___ _______ _________ ________ __________ __________ ______ _______ _______ _____ ___.
_____ ____ _____ ________ ______ _____ _______ __________ __________.
_____ _________ _______ __________ ____ ___ __________ ____ _________ _____.
__________ _______ ____ _____ ________ ______ _____ ____.
__________ ______ _______ _________ ____ ___ ____ ___ ______ __________ _____ ___.
__________ ______ ____ ___ _______ __________ ____ ___ __________.
______ _______ ____ ____ ____ ________ ____ ______ ______ ___ _________.
_____ __________ __________ ________ ________ ___ ______ ______ ______ ______ _______.
_______ ______ ____ ________ _________ _________.
_______ ________ _______ ____ ______ ____ ______ ___ _______ __________ __________.
________ _____ ________ ___ ___ ______ ___ __________ _______ ___ __________ ________.
_______ _______.
Get Full Answer on WhatsApp
State two real-world problems where you think that mathematical modelling is the only approach to find the solution of the problem. Give 4 essentials for each of the problems. Why do you think that there is no other scientific alternative for the treatment of these problems.
b) Classify the following into linear and non-linear models, justifying your classification.
i) Simple harmonic motion for small amplitude of oscillation.
ii) Population growth model given by
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
$
and pt is the price in period t.
यदि एक ग्रह अपनी कक्षा में, जिसे वृत्ताकार माना गया है, अचानक रूक जाता हो, तो दिखाइए कि वह उस कालावधि में सूर्य में गिर जाएगा जो कि ग्रह के परिक्रमण काल का गुना है। √2 8
एक दिए हुए वर्ष में पैदा हुए व्यक्तियों का एक समूह लीजिए और मान लीजिए इस समूह में से t वर्ष बाद जीवित रहने वाले व्यक्तियों की संख्या n(t) है। मान लीजिए x(t) इस समूह के उन सदस्यों की संख्या है, जिन्हें t वर्ष बाद तक चेचक नहीं हुआ है और इसलिए वे अभी तक सुग्राह्य बने हुए हैं। मान लीजिए
वह दर है जिससे सुग्राह्य व्यक्तियों को चेचक होता है और
वह दर है जिससे चेचक से पीड़ित व्यक्तियों की मृत्यु हो जाती है। अंत में मान लीजिए कि चेचक के अतिरिक्त अन्य कारणों से व्यक्ति की मृत्यु होने की दर
है। यदि dx/dt और dn/dt क्रमशः वह दर हो जिससे चेचक होने के कारण तथा अन्य कारणों से मृत्यु होने पर सुग्राह्य व्यक्तियों की संख्या और पूरी जनसंख्या में कमी आती हो, तो
i) dx/dt और dn/dt के समीकरण लिखकर ऊपर की समस्या का सूत्रण करें।
ii) लेकर यह दिखाइए कि z निम्नलिखित आदिमान समस्या को संतुष्ट करता है।
$
iii) किसी भी समय t पर z(t) का मान ज्ञात कीजिए।
iv) बर्नाली ने यह आकलित किया था कि इन मानों की सहायता से 20 वर्ष की आयु वाले उन व्यक्तियों का समानुपात ज्ञात कीजिए जिन्हें चेचक नहीं हुआ था।
क) सरल आवर्त गति में गतिमान कण के वेग 8cm/sec और 6cm/sec होते हैं जबकि वह अपने गति-केन्द्र से क्रमशः 3cm और 4cm की दूरी पर होता है। गति का आवर्त काल और आयाम ज्ञात कीजिए।
A parachutist, whose weight (actually mass) is , drops from a helicopter
above the ground. She falls towards the earth under the influence of gravity. Assume that the gravitational force is constant. Assume that the force due to air resistance is proportional to the velocity of the parachutist. The proportionality constant is
when the parachute is closed, and is
when it is open. If the parachute does not open until
after the parachutist leaves the helicopter, after how many seconds will she hit the ground?
b) A projectile is fixed with a constant speed v at two different angles of projection and
such that it gives the same range. Show that
ompare the phase diagrams of the systems:
i)
ii)
by locating the equilibrium points and sketching the phase paths.
b) Consider the epidemic model governed by the following equation
with initial condition at
. Here x(t) is the number of susceptibles at time t,
is the contact rate. The population is assumed to be closed and homogeneously mixing. Let the contact rate be 0.002 and the number of susceptibles be 5000 initially
i) Find the density of the population when the rate of appearance of new cases is maximum.
ii) Find the time (in weeks) at which the rate of appearance of new cases is maximum.
iii) Obtain the maximum rate of appearance of new cases.
Consider a one-dimensional growth c(x, t) of phytoplankton in a water mass. Formulate the model describing the dynamics of growth taking into account the following: D, its diffusion coefficient, r its rate of growth, R its mortality rate due to sinking. Fixing the area of interest as and the initial concentration of phytoplankton as
, find the concentration distribution of phytoplankton in
at any time t.
b) When an aeroplane ascends from take-off to an altitude of , by how much does the gravitational attraction acting on it decrease?