A rock with mass m=3.00 kg falls from rest in a viscous...

A rock with mass $m=3.00$ kg falls from rest in a viscous medium. The rock is acted on by a net constant downward force of 18.0 $\mathrm{N}$ (a combination of gravity and the buoyant force exerted by the medium) and by a fluid resistance force $f=k v$ where $v$ is the speed in $\mathrm{m} / \mathrm{s}$ and $k=2.20 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}($ see Section 5.3$) .$ (a) Find the initial acceleration $a_{0 .}$ (b) Find the acceleration when the speed is 3.00 $\mathrm{m} / \mathrm{s} .$ (c) Find the speed when the acceleration equals 0.1$a_{0}$ (d) Find the terminal speed $v_{\mathrm{t}}$ . (e) Find the coordinate, speed, and acceleration 2.00 s after the start of the motion. (f) Find the time required to reach a speed of 0.9$v_{t} .$

A rock with mass $m=3.00$ kg falls from rest in a viscous medium. The rock is acted on by a net constant downward force of 18.0 $\mathrm{N}$ (a combination of gravity and the buoyant force
exerted by the medium) and by a fluid resistance force $f=k v$ where $v$ is the speed in $\mathrm{m} / \mathrm{s}$ and $k=2.20 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}($ see Section 5.3$) .$ (a) Find the initial acceleration $a_{0 .}$ (b) Find the acceleration when the speed is 3.00 $\mathrm{m} / \mathrm{s} .$ (c) Find the speed when the acceleration equals 0.1$a_{0}$ (d) Find the terminal speed $v_{\mathrm{t}}$ . (e) Find the coordinate, speed, and acceleration 2.00 s after the start of the motion. (f)
Find the time required to reach a speed of 0.9$v_{t} .$

Key Concepts

First-Order Differential Equations in Motion

The motion of an object under a constant driving force and a drag force proportional to its velocity can be described by a first-order linear differential equation. Solving this equation yields an exponential behavior in the velocity function, illustrating how the object accelerates initially and asymptotically approaches terminal velocity over time.

Terminal Velocity

Terminal velocity occurs when the resistive drag force balances the net driving force (such as gravity adjusted for buoyancy), leading to a net force of zero. At this point, the object stops accelerating and continues to travel at a constant speed.

Exponential Approach to Equilibrium

Because the drag force introduces a damping effect, the object's velocity and acceleration change in an exponential manner as it transitions from its initial state to terminal motion. This exponential approach characterizes the transient dynamics of the system and is important for predicting the time evolution of speed and displacement.

Newton's Second Law

This fundamental principle of mechanics states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. It is used to determine the acceleration when forces such as gravity, buoyancy, and drag are acting on an object.

Viscous Drag Force

In a viscous medium, the resistance (or drag) force typically depends linearly on the object's speed. This drag force, often modeled as proportional to the velocity, opposes the motion and plays a key role in determining both the transient acceleration and the eventual terminal velocity of the object.

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