Straight-Line Motion, Displacement, Velocity & Acceleration Explained | Chapter 2 of University Physics

In Chapter 2 of University Physics, we develop the core kinematic tools needed to describe motion along a single axis. We introduce displacement, average and instantaneous velocity, average and instantaneous acceleration, and learn to interpret x–t, v–t, and a–t graphs. You’ll also see how the four constant-acceleration equations apply to free‐fall problems and how integration generalizes motion analysis when acceleration varies.

Key Kinematic Quantities

• Displacement (∆x) – Vector change in position along the x-axis.
• Average Velocity (v̄ₓ) – ∆x ÷ ∆t, direction-sensitive.
• Instantaneous Velocity (vₓ) – dx/dt; slope of the x–t graph.
• Average Acceleration (āₓ) – ∆vₓ ÷ ∆t.
• Instantaneous Acceleration (aₓ) – dvₓ/dt; slope of the v–t graph.

Graphical Analysis of Motion

Position-time (x–t) graphs reveal velocity as their slope, while velocity-time (v–t) graphs reveal acceleration. Areas under a v–t curve give displacement; areas under an a–t curve give change in velocity. Signs of velocity and acceleration determine whether an object speeds up or slows down.

Constant Acceleration & Kinematic Equations

Under constant acceleration, four equations link x, v, a, and t:

• v = v₀ + a t
• ∆x = v₀ t + ½ a t²
• v² = v₀² + 2 a ∆x
• ∆x = ½ (v₀ + v) t

Use a clear problem-solving strategy: choose your axis, list knowns/unknowns, select the equation that matches, solve algebraically, then check units and physical reasonableness.

Free-Fall Motion

Free fall treats acceleration as constant g = 9.80 m/s² downward. By defining upward as positive, replace aₓ with –g and x with y in kinematic equations to analyze objects in free fall.

Motion with Variable Acceleration (Integration)

When acceleration isn’t constant, integrate a(t) to find v(t) and integrate v(t) to find x(t):

• v(t) = v₀ + ∫₀ᵗ a(t′) dt′
• x(t) = x₀ + ∫₀ᵗ v(t′) dt′

This approach corresponds to areas under acceleration or velocity curves and is essential for more advanced dynamics.

Conclusion

Chapter 2 equips you with the language and tools to analyze one-dimensional motion precisely. Mastering displacement, velocity, acceleration, and their graphical and calculus-based interpretations lays the groundwork for exploring two- and three-dimensional motion in subsequent chapters.

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