Notes on Constant Acceleration and Velocity (Transcript-based)

Transcript Context

• The speaker identifies how the equation sheets are organized for motion problems.
• Specifically mentions: the first three equations are for constant acceleration.
• And for constant velocity, there is only one equation on the sheet.

Constant Acceleration Equations

• With constant acceleration, the standard set of equations (often referred to as SUVAT) appear on the sheet:
  • $s = u t + \tfrac{1}{2} a t^2$
  • $v = u + a t$
  • $v^2 = u^2 + 2 a s$
• Additional related form (useful for certain problems):
  • $s = \frac{u + v}{2}\, t$
• Symbols and meanings:
  • $s$ = displacement (along the motion path)
  • $u$ = initial velocity at time $t=0$
  • $v$ = final velocity at time $t$
  • $a$ = constant acceleration
  • $t$ = time elapsed
• Significance of constant acceleration:
  • Enables direct relationships between displacement, velocity, acceleration, and time without requiring variable acceleration terms.
  • Under the constant-acceleration assumption, the average velocity over the interval is $\bar{v} = \frac{u+v}{2}$, which leads to the alternative displacement form above.

Constant Velocity Equation

• For motion with constant velocity, the equation sheet contains a single equation (since acceleration is zero):
  • Displacement relation:
  • $x = x_0 + v t$
• Variants:
  • If starting from the origin, $x = v t$ when $x_0 = 0$
  • In general, $\Delta x = v \Delta t$ where $\Delta t$ is the elapsed time and $\Delta x = x - x_0$
• Assumptions for this case:
  • The velocity $v$ is constant over the interval, so $a = 0$
  • Motion along a straight line (1D kinematics) unless stated otherwise

Definitions and Variable Roles

• Distinguish between displacement and distance traveled:
  • Displacement $s$ (or $\Delta x$) is the straight-line change in position.
  • Distance traveled may differ from displacement if motion changes direction.
• Initial conditions vs. final conditions:
  • Initial velocity $u$ is at the start time; final velocity $v$ after time $t$
• Roles of time and acceleration:
  • Time $t$ is the independent parameter that links velocity and displacement under constant acceleration.
  • Acceleration $a$ measures the rate of change of velocity; constant in these formulations.

Worked Examples (Illustrative)

• Example 1: Constant acceleration
  • Given: $u = 0\ \text{m/s},\ a = 2\ \text{m/s}^2,\ t = 5\ \text{s}$
  • Displacement: $s = u t + \tfrac{1}{2} a t^2 = 0(5) + \tfrac{1}{2}(2)(5)^2 = 25\ \text{m}$
  • Final velocity: $v = u + a t = 0 + (2)(5) = 10\ \text{m/s}$
• Example 2: Constant velocity
  • Given: $v = 8\ \text{m/s},\ t = 12\ \text{s}$, starting at origin $x_0 = 0$
  • Displacement: $\Delta x = v t = 8\times 12 = 96\ \text{m}$
  • If starting position is not zero: $x = x_0 + v t$

Connections to Foundational Principles

• Relationship to Newtonian mechanics:
  • Constant acceleration problems are 1D kinematics derived under Newton's laws with a constant net force leading to constant acceleration.
• Practical relevance:
  • Useful in problems involving freely falling objects (near Earth's surface with gravity, approximated as constant over short intervals), vehicle acceleration/deceleration, and projectile motion in 1D components.
• Limitations:
  • Real-world motion often involves non-constant acceleration; these equations are approximations valid over chosen time intervals where acceleration can be considered constant.

Notation and Conventions

• Common letters used in kinematics:
  • Position/displacement: $s$ or $x$ or $\Delta x$
  • Velocities: $u$ (initial), $v$ (final)
  • Acceleration: $a$
  • Time: $t$
• Keep track of units to ensure consistency (e.g., meters, seconds).

Practical Tips for Exam Preparation

• Memorize the core SUVAT equations and know when to apply each:
  • Use $s = ut + \tfrac{1}{2} at^2$ when you know initial velocity, acceleration, and time.
  • Use $v = u + at$ when you know initial velocity, acceleration, and time and want final velocity.
  • Use $v^2 = u^2 + 2 a s$ when you know velocities and displacement but not time.
  • Use $s = \frac{u+v}{2} t$ as a quick check when velocity changes uniformly.
• For constant velocity problems, remember the simple relation $\Delta x = v \Delta t$ (or $x = x_0 + v t$).
• Practice with units and check dimensional consistency as a quick verification.