Uniform Accelerated Motion and Kinematic Equations

Conditions of Speeding Up and Speeding Down in a Straight Line

• Speeding Up (Increase in Speed):

  • This occurs when the velocity vector $\mathbf{v}$ and acceleration vector $\mathbf{a}$ are parallel to each other.
  • The angle between velocity and acceleration is $\theta = 0^{\circ}$.
  • Speeding up happens in two sign-based cases:
  • Both velocity and acceleration are positive ($(+v)$ and $(+a)$).
  • Both velocity and acceleration are negative ($(-v)$ and $(-a)$).

• Speeding Down (Decrease in Speed) / Retardation:

  • This occurs when the velocity vector $\mathbf{v}$ and acceleration vector $\mathbf{a}$ are antiparallel to each other.
  • The angle between velocity and acceleration is $\theta = 180^{\circ}$.
  • Speeding down happens when the variables have opposite signs:
  • Velocity is positive and acceleration is negative ($(+v)$ and $(-a)$).
  • Velocity is negative and acceleration is positive ($(-v)$ and $(+a)$).

Uniform Accelerated Motion

• Core Characteristics:
  • The acceleration $\mathbf{a}$ is constant throughout the entire duration of the motion.
  • The change in velocity in equal time intervals is always the same.
  • Constant acceleration can be expressed by the equality of velocity changes over time: $a = \frac{v_2 - v_1}{t_1} = \frac{v_4 - v_3}{t_2}$.

Equations of Motion

These equations are exclusively valid for uniform accelerated motion:

1. Velocity-Time Relation:

   • $\mathbf{v} = \mathbf{u} + \mathbf{a}t$

2. Displacement-Time Relation (Initial Velocity):

   • $\mathbf{s} = \mathbf{u}t + \frac{1}{2}\mathbf{a}t^2$

3. Displacement-Time Relation (Final Velocity):

   • $\mathbf{s} = \mathbf{v}t - \frac{1}{2}\mathbf{a}t^2$

4. Velocity-Displacement Relation:

   • $v^2 = u^2 + 2as$

5. Displacement-Average Velocity Relation:

   • $\mathbf{s} = (\frac{\mathbf{u} + \mathbf{v}}{2})t$

Variable Key:

• $v$: Final velocity
• $u$: Initial velocity
• $a$: Acceleration
• $t$: Time interval
• $s$: Displacement, calculated as the change in position $\mathbf{s} = \mathbf{r}_f - \mathbf{r}_i$

Procedural Guidelines for Motion Problems

• Step 1: Select the origin. Generally, the starting point of the motion is considered the origin.
• Step 2: Select one direction as positive ($+$) and the opposite direction as negative ($-$).
• Step 3: Always input all variables into equations with their proper sign according to the chosen convention.

Special Observations and Ratios

• Galileo's Law of Odd Numbers: If the initial conditions are set such that $v = 0$ (starting from rest) and the acceleration $a$ is constant, the ratio of displacements in successive equal time intervals ($s_1 : s_2 : s_3$) is represented by the ratio of odd numbers:
  • $s_1 : s_2 : s_3 = 1 : 3 : 5$
• Chronological Data:
  • Session: Day-07
  • Date: 24/07/24