Mathematical Modeling (a) Suppose a falling body of mass m experiences gravity,
ID: 2864014 • Letter: M
Question
Mathematical Modeling
(a) Suppose a falling body of mass m experiences gravity, and additionally, an opposing force of air resistance that is proportional to its instantaneous velocity v. Use Newton’s second law to derive a first–order differential equation for the velocity v of the body at time t. Hint: Let k be the proportionality constant relating the air resistance to the velocity.
(b) Two populations, one of predators and one of prey can be modeled by a system of two first–order differential equations. Let x(t) represent the number of predators and y(t) the number of prey. Each equation should be written in normal form, that is with the derivative on the left hand side, and with one positive OR one negative term on the right hand side. 1. A predator population naturally decreases at a rate proportional to its size. 2. A prey population naturally increases at a rate proportional to its size. 3. At time t, the rate of change of one population due to interactions with the other is proportional to the product of the two populations at that time, i.e. x(t)y(t). Interactions are positive for predators and negative for prey. You will need to introduce four proportionality constants; use the symbols a, b, c and d, and assume they are all positive. Assign the proportionality constants however you wish.
(c) Suppose a hole is drilled through the center of the Earth. A body with mass m is dropped into the hole. Let r(t) be the distance from the center of the Earth to the mass at time t. Let M denote the mass of earth, R denote the radius of the earth, and Mr denote the mass of the portion of earth within a sphere of radius r. Assume that the Earth has a constant density ?. Newton’s law of gravitation states that the gravitational force on m, when it is at a distance r(t) from the center of the earth, is given by F = ? kmMr r 2 where the minus sign accounts for the attracting force. Use Newton’s second law to derive the following differential equation: d 2 r dt2 + ? 2 r = 0. where ? 2 = kM R3 .
(a) Suppose a falling body of mass m experiences gravity, and additionally, an opposing force of air resistance that is proportional to its instantaneous velocity v. Use Newton's second law to derive a first-order differential equation for the velocity v of the body at time t. Hint: Let k be the proportionality constant relating the air resistance to the velocity. b) Two populations, one of predators and one of prey can be modeled by a system of two first-order differential equations. Let r(t) represent the number of predators and y(t) the number of prey. Each equation should be written in normal form, that is with the derivative on the left hand side, and with one positive OR one negative term on the right hand side. 1. A predator population naturally decreases at a rate proportional to its size 2. A prey population naturally increases at a rate proportional to its size 3. At time t, the rate of change of one population due to interactions with the other is proportional to the product of the two populations at that time, .e z(t)y(t). Interactions are positive for predators and negative for prey. You will need to introduce four proportionality constants; use the symbols a, b. c and d, and assume they are all positive. Assign the proportionality constants however you wishExplanation / Answer
Newton's Second Law of Motion
a=FmorF=ma=md2rdt2.
F=dpdt,
F=F(r,v,t).
Force Depends on Time: F=F(t)
md2xdt2=F(t).
v(t)=v0+1m0tF()d.
x(t)=x0+0tv()d,
please write other question seperately
Newton's Second Law of Motion
Newton's second law establishes a relationship between the force F acting on a body of mass m and the acceleration a caused by this force.The acceleration a of a body is directly proportional to the acting force F and inversely proportional to its mass m, that is
a=FmorF=ma=md2rdt2.
This formulation is valid for systems with constant mass. When the mass changes (for example, in the case of relativistic motion), Newton's second law takes the formF=dpdt,
where p is the impulse (momentum) of the body.In general, the force F can depend on the coordinates of the body, i.e., the radius vector r, its velocity v, and time t:
F=F(r,v,t).
Below we consider the special cases where the force F depends only on one of these variables.Force Depends on Time: F=F(t)
Assuming that the motion is one-dimensional, Newton's second law is written as the second order differential equation:md2xdt2=F(t).
Integrating once, we find the velocity of the body v(t):v(t)=v0+1m0tF()d.
Here we assume that the body begins to move at time t=0 with the initial velocity v(t=0)=v0. Integrating again, we get the law of motion x(t):x(t)=x0+0tv()d,
where x0 is the initial coordinate of the body, is the variable of integration.