[SOLVED] MA508 Homework 1

1. Consider the following chemical reaction, where one chemical (A) turns into a different chemical (B) and vice versa. Suppose that the total amount of chemical is constant, that is A(t) + B(t) = C, where C is a positive constant. This reaction can be represented schematically in the following way:

where the two positive constants k⁺ and k⁻ are called rate constants.

The following differential equation describes how A changes with time

(1) Recall that, in addition to this differential equation, we also have the conservation constraint A(t)+B(t) = C.

1. Solve for A(t), given A(0) = A₀, with A₀ being a positive constant such that A₀ < C.
2. Use Matlab to check your answer for a few choices of A₀, C, k⁺, and k⁻. (I have provided code that will assist you).
3. The position of an object moving in 1D (x(t)) on a damped, linear spring obeys the following differential equation

mx¨ = −bx˙ − kx                                                                                    (2)

where m, b, and k are positive constant representing the mass of the object, the damping coefficient and the stiffness of the spring, respectively.

1. Solve for x(t), given x(0) = x₀, and ˙x(0) = v₀.
2. Use Matlab to check your answer for a few choices of x₀, v₀, m, b, and k. (I have provided code that will assist you).
3. The following equation describes the velocity, v(t), of a relatively large object falling through a relatively inviscid medium (e.g., a baseball falling through the air)

mv˙ = −cv|v| + mg                                                                                  (3)

where m, c and g are positive constants representing the mass of the object, the drag of the medium, and the pull of gravity. a) draw a plot of ˙v vs. v. Label any equilibrium point(s) and indicate the stability of each. On the horizontal axis, indicate the flow direction.

1. without solving the equation, sketch v(t) as a function of t for several different initial conditions.
2. solve the equation for v(t), given v(0) = 0. (It will simplify your life to assume that v ≥ 0 to get rid of the absolute value sign. Once you have a solution, you can determine whether this is a reasonable assumption).
3. Use Matlab to check your solution. I have not provided code, but you should be able to modify the codefor problem 1.
4. Turn in a completed version of worksheet 1, which you worked on during class on September 1.