

The laws of motion are often far more complicated than the simple law of gravity.
The Force of GravityConstant gravitational acceleration was established by Galileo, but Newton provided the more general form. Acceleration and ForceRecall the law of gravity near the Earth’s surface, describing the position of an object travelling vertically: y = y_{0} + v_{y0}t − ½ gt^{2} If we take the derivative of this expression we get one for the velocity: v = dy/dt = v_{y0} − gt which shows how it becomes more and more negative with time in response to the downward acceleration of gravity. If we take another derivative we get an expression for the acceleration: a = dv/dt = −g We can write this in vector notation using bold face to represent vector quantities, for example g = (0, −g) since it’s a constant value pointing downward: a = dv/dt = g Experimental examination of gravity and other forces in nature such as friction tell us that the important kinematic quantity here is the vector acceleration, which is zero except in the presence of such forces: m a = Σ_{i} F_{i} where m is the mass of the moving object. This is Newton’s second law of motion. The force of gravity is therefore F_{G} = m g The Force of Air ResistanceAir resistance is a common force that is nonconstant, and works against gravity to slow you down. Air Resistance, AnalyticallyAnother force experienced by an object moving through the air is air resistance, which can be approximated by F_{D} = −½ ρAC_{D}vv The minus sign indicates that the force is opposite to the direction of the velocity; ρ is the density of air, 1.225 kg/m^{3} at sea level, A is the projected area of the object perpendicular to the velocity (i.e. πr^{2} for a sphere), and C_{D} is a dimensionless quantity known as the the drag coefficient. An object traveling through the air therefore has an acceleration given by m a = m g − ½ ρAC_{D}vv From this expression we can see that a larger crosssectional area increases the force of air resistance, so that a feather will drop more slowly than a hammer — but not on the Moon, where there is no air! At low speeds C_{D} ~ b/v, so the magnitude of the force varies directly with speed, while at high speeds C_{D} is approximately constant (in the range of 0.1 – 1) and the magnitude of the force varies as the square of the speed. The former is the result of the displacement of air mass by the object with laminar (smooth) flow of air around it; the latter reflects turbulent flow produced by friction around the ball. A ball thrown through the air, which has relatively low speed, therefore has an acceleration given by dv/dt = g − ½ (ρAb/m)v The factor −½ (ρAb/m) has units of inverse time, so we can define a characteristic time: τ = 2m/(ρAb) and dv/dt = g − v/τ This differential equation can be integrated relatively easily: ∫_{v0}^{v} dv/(g − v/τ) = ∫_{0}^{t} dt −τ ln(v − gτ)_{v0}^{v} = t v = gτ + (v_{0} − gτ)e^{t/τ} The transient exponential term will go away in several characteristic times, and the ball will eventually reach a terminal velocity vf = gτ. The greater the density of air and/or the projected area of the ball, the smaller the terminal velocity. Conversely, the greater the ball’s mass the greater its terminal velocity. A typical value of b = 17 m/s, and for a baseball, A = π (3.7 cm)^{2} = 0.0043 m^{2}, m = 0.15 kg, then τ = 3.4 s, so the terminal velocity is –33 m/s. For a human in free fall, the terminal speed will be 50 m/s to 90 m/s, depending on how they hold themselves (flat or bulletlike). We can now define a drag velocity function for any given time and initial velocity: import numpy as np def vd(t, v0): As usual, there’re a few things going on in this function:
np.array( (0, 1, 2) )/tau ⇒ array([ 0. , 0.29411765, 0.58823529]) Such elementbyelement operations on an array are known as broadcasting. np.exp( np.array( (0, 1, 2) )/tau ) ⇒ array([ 1. , 1.34194177, 1.80080771]) Note: there’s also an An array of time values can be generated with t = np.arange(0,20,0.5) This function is like the builtin v0 = (10, 15) The result is a triplet of arrays print(vdt) ⇒ And we’ll plot this to see what it looks like: def vdplot(vdt): vdplot(vdt) There are new python and
Air Resistance, NumericallyNumerical integration of a differential equation is also possible with the SciPy package: from scipy.integrate import quad Here we’re importing the function In this case we are using the integrand in the integral equation above to calculate the time t corresponding to each speed v in the y direction (Remember that in one dimension v becomes v, a signed value that is negative when an object is falling): ∫_{v0}^{v} dv/(g − v/τ) = ∫_{0}^{t} dt = t Coded in a function: g = 9.81 The keyword We want to integrate this function between vs = vf * np.arange(v0[1]/vf, 1, 0.05) It’s helpful to plot this function to get an idea of its behavior: import matplotlib.pyplot as plt Although the integrand is negative, integrating from a positive The function events = [ quad(integrand, v0[1], v) + (v,) for v in vs ] The returned values have the form for event in events: print('%5.2f %5.2e %6.2f' % tuple(event)) ⇒ We can see the error is very small so we can be confident in the result. Comparing to the result from the exact solution which is calculated for specific times: # convert previous result to array and grab first column of times
1.26 6.90 0.00 The velocities in the third columns match! Remember, in the first case they were input to a calculation, in the second they were output from a calculation. What’s the point other than confirming that integration by quadrature works? Well, now we can use the more general formula for drag which is important for large velocities: m a = m g − ½ ρAC_{D}vv dv/dt = g − (1 + v/c)v/τ ∫_{v0}^{v} dv/(g − (1 + v/c)v/τ) = ∫_{0}^{t} dt = t The constant c = b / C_{D} is very large; for b = 17 m/s and C_{D} = 0.5, c = 34 m/s. This is comparable to the terminal velocity, ~gτ ~ 33 m/s, so the extra term becomes significant on that order of speed. Problem 4: A Situation That Requires Numerical Integration!Calculate the more general solution for m a = m g − ½ ρAC_{D}vv Hint: Think about the effect of v/c both as the ball goes up (



