Position, Velocity, and Acceleration Vectors

Introduction

Kinematics describes the motion of objects without reference to the forces causing that motion. The fundamental vectors—position (r), velocity (v), and acceleration (a)—completely characterize particle motion in space. Understanding the relationships between these quantities through differentiation and integration is essential for analyzing all mechanical systems.

Position Vector

The position vector r(t) describes the location of a particle relative to a chosen origin at time t. In Cartesian coordinates: r(t) = x(t)\(\hat{\imath}\) + y(t)\(\hat{\jmath}\) + z(t)\(\hat{k}\). The magnitude |r| = √(x² + y² + z²) gives the distance from the origin. The path traced by the tip of r(t) as time progresses is called the trajectory. Position is always measured relative to a reference point; there is no absolute position.

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Introduction

Kinematics describes the motion of objects without reference to the forces causing that motion. The fundamental vectors—position (r), velocity (v), and acceleration (a)—completely characterize particle motion in space. Understanding the relationships between these quantities through differentiation and integration is essential for analyzing all mechanical systems.

Position Vector

The position vector r(t) describes the location of a particle relative to a chosen origin at time t. In Cartesian coordinates: r(t) = x(t)\(\hat{\imath}\) + y(t)\(\hat{\jmath}\) + z(t)\(\hat{k}\). The magnitude |r| = √(x² + y² + z²) gives the distance from the origin. The path traced by the tip of r(t) as time progresses is called the trajectory. Position is always measured relative to a reference point; there is no absolute position.

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Velocity Vector

Velocity is the time rate of change of position: v = dr/dt. In component form: v = (dx/dt)\(\hat{\imath}\) + (dy/dt)\(\hat{\jmath}\) + (dz/dt)\(\hat{k}\) = v_x\(\hat{\imath}\) + v_y\(\hat{\jmath}\) + vz\(\hat{k}\). Velocity is tangent to the trajectory at every point. The speed is the magnitude of velocity: |v| = √(v_x² + v_y² + vz²). Instantaneous velocity gives the direction and rate of motion at a specific moment. Average velocity over interval Δt: v_avg = Δr/Δt.

Acceleration Vector

Acceleration is the time rate of change of velocity: a = dv/dt = d²r/dt². Components: a_x = dv_x/dt = d²x/dt², similarly for y and z. Acceleration can change both the magnitude (speed) and direction of velocity. For curved motion, even constant speed produces acceleration (centripetal) due to changing direction. Average acceleration: a_avg = Δv/Δt. Instantaneous acceleration is found by taking the limit as Δt → 0.

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Relations Through Calculus

Given position r(t), differentiate to get velocity v(t), differentiate again for acceleration a(t). Conversely, given acceleration a(t) and initial conditions, integrate to find velocity, integrate again for position. With constant acceleration a: v = v₀ + at, r = r₀ + v₀t + 1/2at². These relationships form the foundation of kinematic analysis.

Graphical Interpretation

Position-time graph slope gives velocity. Velocity-time graph slope gives acceleration; area under curve gives displacement. Acceleration-time graph area gives change in velocity. For 1D motion, the sign of velocity indicates direction (positive/negative axis direction). Zero velocity (slope zero on x-t graph) indicates turning points where direction reverses.

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Solved Example: Finding Position from Velocity

A particle moves with velocity v(t) = (3t² - 2t) m/s along x-axis. At t = 0, x = 2 m. Find position at t = 3 s. Solution: v = dx/dt, so dx = v dt. Integrate: x(t) = ∫(3t² - 2t)dt + C = t³ - t² + C. Using initial condition x(0) = 2: 2 = 0 - 0 + C, so C = 2. Therefore x(t) = t³ - t² + 2. At t = 3: x(3) = 27 - 9 + 2 = 20 m. The particle travels from x = 2 m to x = 20 m in 3 seconds.

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