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 $v(t)$.

A force that opposes the motion of an object, slowing it down. It is velocity-dependent and acts in the opposite direction of motion.

The maximum speed an object reaches when the constant force (like gravity) and the resistive force (like air resistance) balance each other, resulting in zero net acceleration.

The time constant, $\tau = \frac{m}{k}$, indicates how quickly an object reaches terminal velocity. A larger mass or smaller k (weaker resistive force) means a longer time to reach terminal speed.

$k$ is a positive constant that depends on the shape and size of the object and the properties of the medium (like air density). It appears in the resistive force equation: $$\vec{F}_{r} = -k \vec{v}$$

At terminal velocity, the net force is zero, which means the acceleration is also zero. The object stops speeding up.

The resistive force increases.

The object stops accelerating and reaches terminal velocity.

The terminal velocity increases.

The object reaches terminal velocity and stops speeding up.

Air resistance increases.