Set up the differential equation for each problem and use Eulerâs method to solve each equation (include the table of values). Use a step size of ât = 1 second. You may use Excel or similar technology to construct your table. State your solution and answer all parts.
a) An object in free-fall has an acceleration of 32.2 ft/s2 downward plus air resistance upward. Air resistance is proportional to the velocity squared and opposes the motion (and so is upward). Assume the initial velocity is 0 ft/s and the proportionality constant for air resistance is 0.0082. (A) State the differential equation with initial value that can be used to model the velocity of the object at any time. (B) Show a table for Eulerâs Method for the first 10 values. (C) Determine and state the terminal velocity, if one exists.
b) Suppose the air resistance in exercise above is proportional to only the velocity instead of the velocity squared. The air resistance is still assumed to be directed upward opposing the motion. Use the same proportionality constant of 0.0082 for the air resistance and assume an initial velocity of 0 ft/s. (A) State the differential equation with initial value that can be used to model the velocity of the object at any time. (B) Show a table for Eulerâs Method for the first 10 values. (C) Determine and state the terminal velocity, if one exists. (Hint: It could take some time to come close to the terminal velocity.)