Kinematics: Uniform and Nonuniform Motion – Graphs, Equations, and Interpretations

Uniform vs Nonuniform Motion

• Velocity is the slope of the position-time graph: velocity $v = \frac{dx}{dt}$
• Acceleration is the rate of change of velocity: $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$
• Uniform (constant) motion means:
  • Velocity is constant and positive (or negative).
  • Position-time graph is a straight line with slope equal to the velocity.
  • Acceleration is zero.
  • Example interpretation from the transcript:
  • A line with positive slope corresponds to motion in the positive direction; as time increases, position increases.
  • If the object starts on one side of the origin and moves toward the origin and then past it, the velocity remains constant (positive), so acceleration is zero.
  • The velocity being the slope confirms constant velocity; the graph does not “curve.”
• Nonuniform motion means velocity is not constant:
  • Position-time graph is curved (typically quadratic) if acceleration is constant and nonzero.
  • Velocity-time graph is a straight line if acceleration is constant (slope of v-t equals a).
  • Acceleration is not zero when velocity changes.
  • If acceleration is constant and nonzero, velocity changes linearly with time and position changes quadratically with time.
  • If velocity changes but acceleration itself varies with time, both v-t and x-t can take other shapes, but the key point is velocity is not constant.

Graphs and their interpretations

• Position vs. Time (x vs t):
  • Uniform motion: straight line; x(t) = x0 + v t (with v constant).
  • Nonuniform motion with constant acceleration: parabola; x(t) is quadratic in time.
  • The sign of the acceleration affects the curvature direction:
  • a > 0: parabola opens upward; minimum of x(t) occurs where velocity crosses zero.
  • a < 0: parabola opens downward; maximum of x(t) occurs where velocity crosses zero.
• Velocity vs. Time (v vs t):
  • Uniform motion (a = 0): horizontal line; velocity constant.
  • Nonuniform motion with constant acceleration: straight line with slope a (v = v0 + a t).
  • The slope of v-t is the acceleration: positive slope => a > 0; negative slope => a < 0.
• Acceleration vs. Time (a vs t):
  • Uniform motion: a(t) = 0 (horizontal line at zero).
  • Constant acceleration: horizontal line at a ≠ 0 (a is constant).
• Reading information from graphs (summary):
  • If x-t is a straight line with positive slope: uniform motion in the positive direction; v is positive and constant; a = 0.
  • If x-t is linear with negative slope: uniform motion in the negative direction; v is negative and constant; a = 0.
  • If x-t is a parabola opening upward: a > 0; velocity increases over time in the positive direction; x(t) has a minimum.
  • If x-t is a parabola opening downward: a < 0; velocity decreases and may reverse direction; x(t) has a maximum.
  • If v-t is a line with positive slope: a > 0; velocity increases with time.
  • If v-t is a line with negative slope: a < 0; velocity decreases with time.

Key equations (constant acceleration regime)

• Uniform motion (no acceleration):

  • Position: $x(t) = x_0 + v t$
  • Velocity: $v = ext{constant}$
  • Acceleration: $a = 0$

• Nonuniform motion with constant acceleration (a ≠ 0 constant):

  • Velocity as a function of time: $v(t) = v_0 + a t$
  • Position as a function of time: $x(t) = x<em>0 + v</em>0 t + frac{1}{2} a t^2$
  • Acceleration is constant: $a(t) = a$

• Average velocity (useful when velocity is changing):

  • General definition: $v_{ ext{avg}} = \frac{ ext{Δ}x}{ ext{Δ}t}$
  • For constant acceleration (linear v-t), the average velocity between initial and final times is the average of initial and final velocities:
  • $v<em>{ ext{avg}} = \frac{v</em>0 + v_f}{2}$

• Relationship between average velocity and displacement for nonuniform motion with varying velocity:

  • In general: $ext{Δ}x = v_{ ext{avg}} imes ext{Δ}t$
  • When velocity varies linearly (constant a): $v<em>{ ext{avg}} = \frac{v</em>0 + v<em>f}{2} ext{ and } ext{Δ}x = v</em>{ ext{avg}} ext{Δ}t = \frac{v<em>0 + v</em>f}{2} ext{Δ}t$

• Kinematic equations (summary of how they interrelate):

  • Velocity as a function of time: $v = v_0 + a t$
  • Position as a function of time: $x = x<em>0 + v</em>0 t + frac{1}{2} a t^2$
  • If you solve for time from the first and substitute, you can derive relationships; similarly, from the second you can relate displacements to time when a is known.

• Quadratic form and turning points (in x-t):

  • A general quadratic in time: $x(t) = a t^2 + b t + c$
  • Turning point (where velocity is zero) occurs at $t_{ ext{tp}} = -\frac{b}{2a}$
  • If a > 0, turning point is a minimum; if a < 0, turning point is a maximum.

Interpretations and connections

• How the sign of acceleration relates to motion:
  • Positive acceleration (a > 0) generally corresponds to velocity increasing in the positive direction or velocity increasing in the negative direction depending on the current sign of v; velocity and acceleration signs together determine whether the speed is increasing or decreasing in a given direction.
  • Negative acceleration (a < 0) similarly affects whether the particle speeds up or slows down depending on the current velocity sign.
  • In particular:
  • If velocity is positive and a > 0, the object speeds up to the positive direction.
  • If velocity is negative and a < 0, the object speeds up to the negative direction.
• For nonuniform motion with acceleration changing over time, you still can use v-t and x-t relations, but the simplest common case is constant acceleration so the standard equations apply.
• Real-world relevance: the same equations apply in one dimension and extend to two dimensions (projectile motion) with the same time-based relations, although the x and y components may differ and gravity acts as a constant acceleration in the vertical direction.

Practice scenarios and how to describe them

• Example descriptions from graphs:
  • A velocity-vs-time graph that is a straight line with positive slope: acceleration is positive and constant; velocity increases with time.
  • A velocity-vs-time graph that is a straight line with negative slope: acceleration is negative and constant; velocity decreases with time.
  • A position-vs-time graph that is a parabola opening upward: acceleration is positive; the velocity increases over time; there is a minimum position at the turning point.
  • A position-vs-time graph that is a parabola opening downward: acceleration is negative; the velocity decreases and there is a maximum position at the turning point.
• Describing a given motion from v-t or x-t:
  • If velocity is changing (v-t is not flat), motion is nonuniform.
  • If acceleration is constant and nonzero, you can expect a linear v-t graph and a quadratic x-t graph.
  • If velocity is always negative or always positive but changing in magnitude, describe the direction and magnitude changes separately from the sign of velocity.
• Common confusions to avoid (clarifications):
  • Uniform acceleration means nonuniform motion (the velocity changes while acceleration is constant).
  • The statement that uniform acceleration equals uniform motion is incorrect; uniform acceleration refers to the constant rate of change of velocity, whereas uniform motion refers to constant velocity.

Quick practice prompts (based on the lecture themes)

• Given a velocity-time graph with constant positive slope, describe the motion and the corresponding x(t) and a(t).
• If x(t) is a straight line with positive slope, what can you say about v(t) and a(t)?
• If x(t) is a parabola opening upward, what is the sign of a and what does that imply about the behavior of v(t)?
• If v(t) is a straight line with negative slope, what is the acceleration and how does velocity change over time?
• For a system with constant acceleration, compute the position after time t given initial position x0, initial velocity v0, and acceleration a: $x(t) = x<em>0 + v</em>0 t + frac{1}{2} a t^2$
• Derive the velocity after time t: $v(t) = v<em>0 + a t$ and show the corresponding kinematic relation for displacement: $ext{Δ}x = v</em>{ ext{avg}} ext{Δ}t$ with $v<em>{ ext{avg}} = \frac{v</em>0 + v_f}{2}$ when acceleration is constant.

Notes on context and scope

• The same one-dimensional equations extend to two dimensions (projectile motion) by treating each coordinate with its own x(t), y(t), v(t), and a(t) components while using the same time parameter and gravitational acceleration as appropriate.
• The lecture emphasizes recognizing motion types from graphs and understanding that the slope and curvature encode velocity and acceleration information.
• When reading a graph:
  • Look for straight vs curved x-t: straight implies constant velocity; curved implies changing velocity.
  • Look for straight v-t with slope: constant acceleration; magnitude of slope gives magnitude of acceleration.
  • Look for constant a in a-t: straight horizontal line not at zero means nonzero, constant acceleration.

Summary takeaway

• Uniform motion: x(t) linear, v constant, a = 0.
• Nonuniform motion with constant acceleration: x(t) quadratic, v(t) linear, a constant ≠ 0.
• General kinematic framework connects x, v, and a through the core equations:
  • $v = v_0 + a t$
  • $x = x<em>0 + v</em>0 t + frac{1}{2} a t^2$
  • $a = \frac{ ext{d}v}{ ext{d}t} = \frac{ ext{d}^2x}{ ext{d}t^2}$