(III) The force of air resistance (drag force) on a rapidly...

(III) The force of air resistance (drag force) on a rapidly falling body such as a skydiver has the form $F_{\mathrm{D}}=-k v^{2},$ so that Newton's second law applied to such an object is $$ m \frac{d v}{d t}=m g-k v^{2} $$ where the downward direction is taken to be positive. (a) Use numerical integration [Section $2-9]$ to estimate (within $2 \%$ ) the position, speed, and acceleraton, from $t=0$ up to $t=15.0 \mathrm{~s},$ for a $75-\mathrm{kg}$ skydiver who starts from rest, assuming $k=0.22 \mathrm{~kg} / \mathrm{m}$ (b) Show that the diver eventually reaches a steady speed, the terminal speed, and explain why this happens. (c) How long does it take for the skydiver to reach $99.5 \%$ of the terminal speed?

(III) The force of air resistance (drag force) on a rapidly falling body such as a skydiver has the form $F_{\mathrm{D}}=-k v^{2},$ so that Newton's second law applied to such an object is
$$
m \frac{d v}{d t}=m g-k v^{2}
$$
where the downward direction is taken to be positive.
(a) Use numerical integration [Section $2-9]$ to estimate (within $2 \%$ ) the position, speed, and acceleraton, from $t=0$ up to $t=15.0 \mathrm{~s},$ for a $75-\mathrm{kg}$ skydiver who starts from rest, assuming $k=0.22 \mathrm{~kg} / \mathrm{m}$ (b) Show that the diver eventually reaches a steady speed, the terminal speed, and explain why this happens. (c) How long does it take for the skydiver to reach $99.5 \%$ of the terminal speed?

Key Concepts

Differential Equations in Dynamics

Differential equations in dynamics are used to describe how physical systems evolve over time. In the context of a falling object with air resistance, the equation m dv/dt = mg - kv^2 models how velocity changes due to the combined effects of gravity and drag, and analyzing this equation helps understand the transition to terminal velocity.

Numerical Integration

Numerical integration refers to the techniques used to approximate the solutions of differential equations, particularly when analytical solutions are difficult or impossible to obtain. Methods such as Euler's method or Runge-Kutta techniques help estimate position, speed, and acceleration over time by discretizing the continuous variables.

Newton's Second Law

Newton's Second Law relates the net force acting on an object to its mass and the acceleration it experiences. In problems involving forces such as gravity and drag, this law is used to set up differential equations where the sum of all forces is equal to the mass multiplied by the rate of change of velocity.

Drag Force

Drag force is a resistive force experienced by an object moving through a fluid like air or water. Often modeled as being proportional to the square of the velocity (i.e., F_D = -kv^2), this force opposes the motion of the object and becomes significant at high speeds, affecting the acceleration and overall motion.

Terminal Velocity

Terminal velocity is the steady speed reached by an object when the force of gravity is exactly balanced by the drag force. At this point, the net force is zero, resulting in zero acceleration. This phenomenon explains why, over time, objects subject to air resistance will stop accelerating and continue falling at a constant speed.

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