How would I create and analyze a differential equation to model the dynamics of a falling object that experiences a resistive force F = bv (where v is the object's instantaneous velocity and b is a constant) in the presence of air friction? (The object has mass m, and acceleration due to gravity is g.)
If we consider a free-body diagram of the falling object to show the forces acting on it, we would have the weight (mg) of the object acting downward and the resistive force of quantity bv acting upward that retards its motion. Since acceleration, a, is the first derivative of velocity, then Newton's Second Law would give us F_net = ma = m*(dv/dt). Applying this law and our free-body diagram for this situation, we would have: F_net = mg - bv = m*(dv/dt) [We note the the direction/sign of acceleration corresponds to the direction/sign of the weight vector.] Then we could divide both sides of the differential equation by mass m and collect the velocity terms on one side of the equation, yielding: dv/dt + (b/m)*v = g We would then proceed to separate the variables of velocity and time (i.e., separate the differentials dv and dt) and continue by integrating both sides of the equation. dv / [g - (b/m)*v] = dt
In applying the Laws of Logarithms, why can we state that log_b (A^c) = c*log_b (A) ? (Note: Let us here read log_b (A) as "log base b of A")
Let log_b (A) = u. Then, in exponential form, this would be b^u = A. We then have log_b (A^c) = log_b (b^u)^c = log_b [b^(u*c)] = u*c. Since u = log_b (A) from above, then u*c = c*log_b (A).
Why does a function F have positive values for its second derivative (F'' > 0) over an interval J where F is concave upward?
In studying the nature of functions and their derivatives, we learn that a function has increasing values on intervals where the function's derivative is positive. So, in extending this idea to second derivatives, the second derivative, F'', will be positive when the first derivative, F', is increasing. If F' is increasing on the interval J, this means that the slopes of tangent lines to F are changing from more negative to less negative to zero to less positive to more positive as we move across interval J from left to right. This indicates that F changes from decreasing to increasing (and through a local minimum) over the interval J, giving the graph of F an upward-opening bowl shape (i.e., concave upward) on J.