chapter 2 in person 8-28-25 One-Dimensional Motion: Uniform Motion and Velocity-Time Graphs

Uniform Motion

• Uniform motion: constant velocity in a fixed direction; velocity vector points the same way; speed is constant; position-time graph is a straight line with equal displacements per equal time.

  • This implies zero acceleration. The SI unit for position is meters (m), for time is seconds (s), and for velocity is meters per second (m/s).

• Slope of a position-time graph equals velocity.

• For uniform motion, velocity is constant: there is a direct relation between position change and time.

Equation of Motion for Uniform Motion

• Basic relation: v=\frac{\Delta x}{\Delta t}

• First equation of motion for uniform motion: x_f = x_i + v_i\Delta t

• Here \Delta t = t_f - t_i; for uniform motion this describes how far the object moves in time \Delta t.

Meaning of variables

• x_f: final position

• x_i: initial position

• v_i: initial velocity (along the x-direction)

• \Delta t: elapsed time (time interval)

Simplifications and common conventions

• Often take t_i = 0; thus \Delta t = t_f = t.

• Often take x_i = 0 (origin); then x_f = v_i t.

Velocity-Time Graph and displacement

• The area under a velocity-time graph between times gives displacement: \Delta x = \int_{t_1}^{t_2} v(t)\,dt

• For constant velocity, \Delta x = v\,\Delta t (area of a rectangle).

  • For uniform motion, the velocity-time graph is a horizontal line, reflecting the constant velocity. The area under this horizontal line perfectly matches the v\,\Delta t formula.

• Qualitatively, larger area under the curve means larger displacement.

Instantaneous velocity and slope

• Instantaneous velocity at time t is the slope of the position-time graph at t: v(t) = \frac{dx}{dt}.

• When velocity changes, the instantaneous velocity is best represented by a tangent line (infinitesimally small interval).

Constructing a velocity-time graph from a position-time graph (qualitative)

• Identify key points on the position-time curve (e.g., a, b, c).

• At each point, the instantaneous velocity is the slope of the tangent there.

  • A positive slope on the position-time graph indicates a positive velocity. A negative slope indicates a negative velocity. A zero slope indicates that the object is momentarily at rest. The steepness of the slope corresponds to the magnitude of the velocity – a steeper slope means a greater speed. When the position-time graph is curving, the slope is changing, indicating varying velocity and thus acceleration.

• Example: at c the slope is zero (velocity = 0); at a the slope starts at 0 and increases; at b the slope is maximum (largest velocity) before decreasing again.

• The resulting velocity-time graph is built from the sequence of instantaneous slopes.

Parabolic position-time graphs and constant acceleration

• If the position-time curve is parabolic (typical for constant acceleration), then:

  • Velocity is linear in time: v(t) = v_i + a t

  • Position is quadratic: x(t) = x_i + v_i t + \tfrac{1}{2} a t^2

  • This relationship arises because acceleration is the rate of change of velocity, and velocity is the rate of change of position. Thus, if acceleration is constant, velocity changes linearly, and position changes quadratically with time.

• Therefore, a constant acceleration yields a straight-line velocity-time graph; non-constant acceleration yields a curved velocity-time graph.

Problem-solving mindset

• Start from the core principle: one-dimensional motion.

• Use the minimal, most useful equations and interpret them via the graphs (position-time and velocity-time).

• When uncertain, break problems into small segments and analyze instantaneous quantities via slopes or areas under curves.