Notes on Uniform and Nonuniform (Constant Acceleration) Motion

Uniform Motion

  • Definition: In uniform motion, velocity is constant, so there is no acceleration (a = 0).
  • Fundamental relationships:
    • Velocity is the rate of change of position: v=dxdtv = \frac{dx}{dt}
    • For uniform motion, displacement is proportional to time: x(t)=x0+v tx(t) = x_0 + v\,t
    • If you include the initial time t0: x(t)=x<em>0+v (t−t</em>0)x(t) = x<em>0 + v\,(t - t</em>0), and typically we take (t_0 = 0).
    • Average velocity concept: v<em>avg=ΔxΔtv<em>{avg} = \frac{\Delta x}{\Delta t}. For uniform motion, this equals the constant velocity, so you can drop the "average" qualifier: (v = v{avg}).
  • Example with a constant velocity: (v = 2\ \text{m/s}), (x_0 = 0).
    • Position as a function of time: x(t)=0+2t=2t mx(t) = 0 + 2t = 2t\,\text{m}.
    • Table:
    • At (t=0\ \text{s}): (x=0) m
    • At (t=1\ \text{s}): (x=2) m
    • At (t=2\ \text{s}): (x=4) m
    • At (t=3\ \text{s}): (x=6) m
    • At (t=4\ \text{s}): (x=8) m
    • At (t=5\ \text{s}): (x=10) m
  • Graphical interpretation (position vs time):
    • Position increases linearly with time for uniform motion, producing a straight line.
    • The slope of the x vs t graph is the velocity: \text{slope} = \frac{\Delta x}{\Delta t} = v = 2\ \text{m/s}.$n- Velocity vs Time graph for uniform motion:
    • Velocity is constant, so the v vs t graph is a horizontal line at (v = 2\,\text{m/s}).
  • Acceleration vs Time graph for uniform motion:
    • Acceleration is zero, so the a vs t graph is a horizontal line at (a = 0).
  • Direction/sign convention (example with negative velocity):
    • If velocity is negative (to the left), the x vs t graph is a line with negative slope; the velocity vs time graph is a horizontal line at the negative value; acceleration remains zero.
  • Quick takeaways:
    • Uniform motion ≡ straight-line x(t) with constant slope (v).
    • Slope of x vs t equals velocity: v = \frac{\Delta x}{\Delta t}.
    • For uniform motion, you do not need to use an average velocity formula; the instantaneous velocity is the same as the average velocity over any interval.

Nonuniform Motion (Constant Acceleration)

  • Definition: In nonuniform motion with constant acceleration, the velocity changes at a constant rate: a = constant.
  • Key velocity-time relation:
    • v(t) = v0 + a\,t where (v0) is the initial velocity at (t = 0).
  • Key position-time relation (with quadratic term):
    • x(t) = x0 + v0\,t + \tfrac{1}{2} a\,t^2 where (x_0) is the initial position at (t = 0).
  • Alternative form using average velocity:
    • Average velocity over a time interval of length (t) is
    • v{avg} = \frac{v0 + v}{2} where (v) is the final velocity after time (t).
    • Then
    • x(t) = x0 + v{avg}\, t = x0 + \frac{v0 + v}{2} t.
  • Another common form without explicit time in terms of velocity:
    • Letting final velocity be (v) after time (t), we can write
    • x = x0 + v{avg}\, t = x0 + \frac{v0 + v}{2}\, t
  • Velocity-position relationship without time (v^2 form):
    • v^2 = v0^2 + 2a\,(x - x0)
    • This eliminates explicit time and is useful when time is not known.
  • A standard set of equations of motion (for constant acceleration):
    • Position with time: x = x0 + v0 t + \tfrac{1}{2} a t^2
    • Velocity with time: v = v_0 + a t
    • Position with velocity without time: x = x0 + \frac{v0 + v}{2} t
    • Velocity-position without time: v^2 = v0^2 + 2 a (x - x0)
  • Graphical interpretation for nonuniform motion:
    • Position vs time: generally non-linear (a parabola) when acceleration is nonzero.
    • Velocity vs time: linear relation with slope equal to acceleration, i.e., a straight line: v(t) = v_0 + a t.
    • Acceleration vs time: constant line at (a).
  • If acceleration is zero, nonuniform motion reduces to uniform motion; otherwise, velocity changes linearly with time and position changes quadratically with time.
  • Sign conventions and interpretation:
    • Positive direction typically to the right; negative velocity indicates motion to the left.
    • The slope of the x vs t graph remains the velocity (positive or negative depending on direction).

a. Example construction for a scenario with (v0 = 2\ \text{m/s}), (a = 2\ \text{m/s}^2), (x0 = 0):

  • Velocity over time: (v(t) = 2 + 2t).
  • Position over time: (x(t) = 0 + 2 t + \tfrac{1}{2} (2) t^2 = 2t + t^2).
  • Times and positions: at (t=1\,s), (x=3\,\text{m}); at (t=2\,s), (x=8\,\text{m}); at (t=3\,s), (x=15\,\text{m}).

Equations of Motion: Summary and How to Choose

  • For uniform motion (a = 0):
    • Position: x = x_0 + v t
    • Velocity: constant, v = \text{const}
    • Acceleration: a = 0
    • Graphs: x(t) linear, v(t) horizontal, a(t) = 0
  • For nonuniform motion with constant acceleration (a ≠ 0):
    • Velocity: v = v_0 + a t
    • Position: x = x0 + v0 t + \tfrac{1}{2} a t^2
    • Alternative forms:
    • x = x0 + \frac{v0 + v}{2} t where (v = v_0 + a t)
    • v^2 = v0^2 + 2 a (x - x0)
  • Conceptual check:
    • If x vs t is a straight line, motion is uniform (a = 0).
    • If x vs t is curved (nonlinear), but v(t) is a straight line, motion has constant acceleration (a ≠ 0).
    • The velocity-time graph will be linear and the acceleration-time graph will be constant when acceleration is constant.
    • The sign of the slope and intercepts reflect direction and initial conditions (e.g., starting position not necessarily at the origin).

Graph Interpretation and Problem-Solving Tips

  • How to identify motion type from graphs:
    • x vs t linear => uniform motion (a = 0).
    • x vs t nonlinear (parabolic) => nonuniform motion with a ≠ 0.
    • v vs t linear => acceleration is constant (the slope equals a).
    • a vs t constant (a) => straight horizontal line on acceleration graph.
  • Key checks:
    • If velocity values change by a constant amount per equal time interval (e.g., 2 m/s every 1 s), acceleration is constant and nonuniform motion applies.
    • If velocity is constant, x increases linearly with time and acceleration is zero.
  • Example interpretation from the transcript:
    • For a particle moving with v = 2 m/s from x0 = 0, the x vs t graph is a straight line with slope 2, and the velocity vs time graph is a constant horizontal line at 2 m/s.
    • For a particle with increasing velocity (e.g., 2, 4, 6, 8, … m/s at successive seconds), acceleration is constant and nonzero, resulting in a parabola for x vs t and a straight line for v vs t.

Worked Relationships (Choose Based on Known Quantities)

  • If you know x0, v0, a, and t:
    • Position: x = x0 + v0 t + \tfrac{1}{2} a t^2
    • Velocity: v = v_0 + a t
  • If you know x0, v0, and x (no t):
    • Velocity-position relationship: v^2 = v0^2 + 2 a (x - x0)
  • If you know x0, v0, and final velocity v after time t (no explicit a):
    • Position: x = x0 + \frac{v0 + v}{2} t
    • Alternatively, with time: x = x0 + v0 t + \tfrac{1}{2} a t^2,\quad v = v_0 + a t
  • If you know x0, and the final velocity after some time, you can also use the average velocity:
    • v{avg} = \frac{v0 + v}{2}andthenand thenx = x0 + v{avg} t
  • The one equation without time that helps when time is not known: v^2 = v0^2 + 2 a (x - x0)

Note on Practical Use

  • You do not need to memorize every form blindly; choose the form that uses the quantities you have available.
  • The four common equations of motion under constant acceleration connect position, velocity, time, and acceleration in different ways depending on what is known.
  • Always confirm the sign convention for direction before applying formulas; changing the origin or defining the positive direction will flip signs in velocities and displacements, but the physics remains the same.

Quick Reference: Key Equations (Constant Acceleration)

  • Position as a function of time: x = x0 + v0 t + \tfrac{1}{2} a t^2
  • Velocity as a function of time: v = v_0 + a t
  • Position using average velocity: x = x0 + v{avg} t, \, v{avg} = \frac{v0 + v}{2}
  • Velocity–position relation without time: v^2 = v0^2 + 2 a (x - x0)$$
  • If no time is needed, you can use the squared-velocity form to relate x and v directly.

Connections and Real-World Relevance

  • Uniform motion models constant-speed motion, foundational for understanding straight-line kinematics in physics and engineering.
  • Nonuniform motion with constant acceleration models common real-world scenarios: vehicles accelerating or objects under gravity near the Earth, where acceleration is approximately constant.
  • These equations underpin motion planning in robotics, car safety simulations, and physics education, providing intuitive links between displacement, velocity, and time.

Summary of Concepts

  • Uniform Motion:
    • Constant velocity, zero acceleration.
    • x(t) is linear in t; v is constant; a = 0.
  • Nonuniform Motion (Constant Acceleration):
    • Velocity changes linearly with time; acceleration is constant.
    • x(t) is quadratic in t; v(t) is linear in t; a(t) is constant.
    • Multiple equivalent forms exist for x and v depending on known quantities; one form without time is v^2 = v0^2 + 2 a (x - x0).
  • Graphical Signatures:
    • Uniform: x vs t linear; v vs t constant; a vs t zero.
    • Nonuniform with constant a: x vs t nonlinear (parabolic); v vs t linear; a vs t constant.

Quick Practice Remarks (from lecture style)

  • To decide if a motion is uniform or nonuniform, first inspect the x vs t plot or the v vs t plot.
  • If velocity is constant, you can use x = x_0 + v t directly; otherwise, use the constant-acceleration forms.
  • For an object starting at a non-origin position and moving with a nonzero velocity, the same equations apply with nonzero x0 and/or v0, and the graphs shift accordingly.