Dynamics: Step-by-Step Derivations

Abstract

This article develops the foundational kinematic relationships in particle dynamics through systematic differentiation and integration. We derive velocity from position and acceleration from velocity, then demonstrate how these relationships enable prediction of particle motion. A worked example illustrates the practical application of these derivations to a concrete motion problem.

Background

Dynamics concerns the motion of bodies under the influence of forces. Before addressing forces, we must establish the kinematic framework—the mathematical description of motion itself ^{[acceleration]}. This framework rests on three linked quantities: position, velocity, and acceleration, each related to the others through calculus.

Position $s(t)$ specifies where a particle is located along a line at time $t$. Velocity describes how quickly position changes. Acceleration describes how quickly velocity changes. These relationships are not independent; they form a chain of derivatives that allows us to move between representations of motion.

Key Results

Deriving Velocity from Position

The velocity function is the time derivative of the position function ^{[velocity-function]}:

$v(t) = \frac{ds}{dt}$

This definition captures the instantaneous rate of change of position. If position is given by a polynomial or other smooth function, differentiation yields velocity as a function of time.

Interpretation: A positive velocity indicates motion in the positive direction; negative velocity indicates motion in the negative direction. When $v(t) = 0$, the particle momentarily stops—a critical point for analyzing changes in direction and total distance traveled ^{[velocity-function]}.

Deriving Acceleration from Velocity

Similarly, acceleration is the time derivative of velocity ^{[acceleration]}:

$a(t) = \frac{dv}{dt}$

Acceleration quantifies how quickly the particle's speed or direction is changing. It can also be expressed as the second derivative of position:

$a(t) = \frac{d^2s}{dt^2}$

This double derivative reveals the deep connection: acceleration measures the curvature of the position-time graph ^{[acceleration-function]}.

Average Velocity

For a finite time interval, we often compute average velocity rather than instantaneous velocity ^{[average-velocity]}:

$v_{\text{avg}} = \frac{\Delta s}{\Delta t}$

where $\Delta s$ is the change in position and $\Delta t$ is the change in time. Average velocity smooths out variations in instantaneous velocity over the interval, providing a summary measure of overall motion ^{[average-velocity]}.

Worked Example

Consider a particle whose position is given by:

$s(t) = 10t^2 + 20 \text{ (mm)}$

We will derive velocity and acceleration, then analyze the motion.

Step 1: Find the velocity function.

Differentiate position with respect to time:

$v(t) = \frac{ds}{dt} = \frac{d}{dt}(10t^2 + 20) = 20t \text{ (mm/s)}$

Step 2: Find the acceleration function.

Differentiate velocity with respect to time:

$a(t) = \frac{dv}{dt} = \frac{d}{dt}(20t) = 20 \text{ (mm/s}^2\text{)}$

In this example, acceleration is constant—the particle experiences uniform acceleration.

Step 3: Analyze the motion at a specific time.

At $t = 2$ s:

• Position: $s(2) = 10(2)^2 + 20 = 40 + 20 = 60$ mm
• Velocity: $v(2) = 20(2) = 40$ mm/s
• Acceleration: $a(2) = 20$ mm/s²

The particle is at 60 mm, moving in the positive direction at 40 mm/s, and speeding up at a constant rate.

Step 4: Compute average velocity over an interval.

From $t = 0$ to $t = 2$ s:

• $\Delta s = s(2) - s(0) = 60 - 20 = 40$ mm
• $\Delta t = 2 - 0 = 2$ s
• $v_{\text{avg}} = \frac{40}{2} = 20$ mm/s

The average velocity (20 mm/s) differs from the instantaneous velocity at $t = 2$ s (40 mm/s) because the particle is accelerating. At $t = 0$, the instantaneous velocity is 0 mm/s; by $t = 2$ s, it has increased to 40 mm/s. The average reflects the overall displacement divided by elapsed time.

References

• ^{[acceleration]}
• ^{[velocity-function]}
• ^{[position-function]}
• ^{[acceleration-function]}
• ^{[average-velocity]}

AI Disclosure

This article was drafted with the assistance of an AI language model. The mathematical derivations, conceptual explanations, and worked example were generated based on the provided class notes and are intended to reflect standard undergraduate dynamics pedagogy. All claims are grounded in the cited notes. The author reviewed the output for technical accuracy and clarity before publication.