1. Apply Newton's Second Law ((F_{net} = ma)). 2. Set up the differential equation for velocity. 3. Use separation of variables. 4. Integrate to find the velocity function. 5. Apply initial conditions to solve for constants.

1. Set the net force equal to zero (resistive force + gravitational force = 0). 2. Solve for velocity ((v_t)), which represents terminal velocity.

1. Velocity approaches terminal velocity (constant value). 2. Acceleration approaches zero. 3. Net force approaches zero.

1. Calculate the time constant, $\tau = \frac{m}{k}$. 2. A larger mass or smaller k (weaker resistive force) means a longer time to reach terminal speed.

1. Draw a free-body diagram to identify all the forces acting on the object. 2. Write out the differential equation and then solve it step-by-step.

Without resistive forces: Constant acceleration (often due to gravity), linear change in velocity. | With resistive forces: Acceleration is not constant, velocity approaches a terminal value, exponential change in velocity.

Larger time constant: Object takes longer to reach terminal velocity, either due to larger mass or smaller k. | Smaller time constant: Object reaches terminal velocity more quickly, due to smaller mass or larger k.

Spread-eagle: Larger surface area, greater air resistance, lower terminal velocity. | Streamlined dive: Smaller surface area, less air resistance, higher terminal velocity.

Feather: Lower mass, higher surface area to mass ratio, reaches terminal velocity quickly, falls slowly. | Bowling ball: Higher mass, lower surface area to mass ratio, takes longer to reach terminal velocity, falls faster.

Initial Velocity: The velocity of an object at the start of its motion, determined by external factors (e.g., how hard it was thrown). | Terminal Velocity: The constant speed that the object eventually reaches when the force of gravity is balanced by the drag force.

1. Apply Newton's Second Law ($\vec{F}_{net} = m\vec{a}$) to obtain a differential equation for velocity.
2. Use separation of variables to solve the differential equation by integrating with appropriate limits.
3. Use initial conditions to find the constants of integration.
4. Solve for the velocity as a function of time.

1. Set the derivative of velocity with respect to time equal to zero: $$frac{dv}{dt} = 0$$.
2. Solve the resulting equation for velocity. This value is the terminal velocity.

1. Identify all forces acting on the object, including resistive forces.
2. Apply Newton's Second Law to set up the differential equation.
3. Solve the differential equation using separation of variables and initial conditions.
4. Analyze the resulting velocity, position, and acceleration functions, paying attention to asymptotes and the time constant.

1. Identify the mass $m$ of the object.
2. Identify the constant $k$ in the resistive force equation ($F_r = -kv$).
3. Calculate the time constant using the formula: $\tau = \frac{m}{k}$.

1. Draw a free-body diagram showing gravitational force and air resistance.
2. Apply Newton's second law: $mg - kv = ma$.
3. Determine the terminal velocity by setting $a = 0$, so $v_t = mg/k$.
4. If required, solve the differential equation for $v(t)$.