Motion in a Straight Line – Comprehensive Notes

Motion in a Straight Line – Comprehensive Study Notes These notes consolidate the key concepts, relationships, graphs, and worked ideas from the provided transcript on motion in one dimension, including graphs, equations of motion, and common problems. Equations are presented in LaTeX format as requested. ---

Kinematics: Graphs - Purpose of graphs:

• x–t (displacement–time) graph: slope gives velocity; area under v gives displacement.
• v–t (velocity–time) graph: slope gives acceleration; area under v gives displacement (signed).
• a–t (acceleration–time) graph: slope relates to jerk (if defined); area under a gives velocity change.
• Key relationships:
• Velocity is the rate of change of position: $v=\frac{dx}{dt}$
• Acceleration is the rate of change of velocity: $a=\frac{dv}{dt}$
• Displacement over a time interval is the signed area under the velocity–time curve over that interval.
• Slope of the displacement–time curve at a point equals the instantaneous velocity at that time: $\text{slope of } x(t) = v(t)$
• Graph-conversion rule of thumb:
• For x–t: slope = velocity; curvature indicates changing velocity.
• For v–t: slope = acceleration; area under curve = displacement during the interval.
• For a–t: area under curve = change in velocity; the sign of a determines whether velocity increases or decreases. ---

Distinct Concepts: Distance vs Displacement - Distance (scalar): total length of the path traveled; always non-negative.

• Displacement (vector): straight-line difference between final and initial position; can be positive, negative, or zero depending on chosen axis.
• In 1D motion along a straight line:
• Distance accumulates as the integral of the absolute value of velocity over time.
• Displacement is the net change in position: $\text{Displacement} = x(t<em>f) - x(t</em>i)$
• Important notes:
• Two different paths can yield the same displacement but different distances traveled.
• Sign conventions matter: positive direction is chosen (e.g., +x to the right, +t forward in time). ---

Equations of Motion (Constant Acceleration) - When acceleration is constant (a constant):

• Velocity: $v = v_0 + a t$
• Displacement: $x = x<em>0 + v</em>0 t + \frac{1}{2} a t^2$
• Velocity–position relation (energy form):$v^2 = v<em>0^2 + 2 a (x - x</em>0)$
• These are the standard equations used for problems with uniform acceleration (e.g., free fall with constant gravity, projectiles in 1D, etc.).
• Special case (initial conditions at origin): if $x<em>0=0, v</em>0=0$, then $x= \frac{1}{2} a t^2,\ v=a t.$ ---

Acceleration as a Function of Time, Position, or Velocity Often a is not constant; three common dependencies and how to handle them:

• Case (i) a = a(t) (depends only on time)
  • Integrate directly to find velocity: $v(t) = v<em>0 + \int</em>{t_0}^{t} a(t') dt$
  • Then integrate the velocity to find position: $x(t) = x<em>0 + \int</em>{t_0}^{t} v(t') dt'$
• Case (ii) a = a(v) (depends only on velocity)
  • Use the definition $a = \frac{dv}{dt}$. Rearrange and integrate: $\int<em>{v</em>0}^{v} \frac{dv'}{a(v')} = \int<em>{t</em>0}^{t} dt' = t - t_0$
  • If position is also needed, use $a = v \frac{dv}{dx}$. Rearrange and integrate: $\int<em>{v</em>0}^{v} \frac{v' dv'}{a(v')} = \int<em>{x</em>0}^{x} dx' = x - x_0$
• Case (iii) a = a(x) (depends only on position)
  • Use the chain rule to write acceleration as a function of x and v: $$a = \frac{dv}{dt} = \frac{dv}{dx} \frac{dx}{dt} =