Physics Kinematics and Dynamics

Distance and Vectors

• Distance:
  • A scalar quantity; measures how much ground an object has covered during its motion.
  • Does not provide information about direction.
• Vectors:
  • Quantities that include both magnitude and direction.
  • Represented with positive (+) or negative (-) signs for direction.

Speed and Velocity

• Speed:
  • Given by the formula:
    $ext{Speed} = \frac{\text{Distance}}{\text{Time}}$
• Average Velocity:
  • Defined as the total displacement divided by total time.
  • Formula:
    $\text{Average Velocity} = \frac{\text{Distance}}{\text{Time}}$
• Instantaneous Velocity:
  • The velocity of an object at a specific moment in time.
  • Different from average velocity.

Key Formulas

• Final Velocity:
  • $V<em>f = V</em>i + a \cdot t$
• Acceleration:
  • $a = \frac{V<em>f - V</em>i}{t}$
• Kinematic Equations:
  1. $2a d = V<em>f^2 - V</em>i^2$
  2. $d = V_i \cdot t + \frac{1}{2} a t^2$

Problem-Solving Strategy

• Givens:
  • Identify what values are provided in a problem.
  • Watch for phrases indicating initial or final velocities (e.g., "starts at rest" implies initial velocity is 0).
• Using Formulas:
  • Identify which formula fits the known variables.
  • You may need to compute intermediate variables if all four variables are not known.

Graphing Principles

• Distance vs. Time Graph:
  • Slope indicates velocity.
  • Steeper slope = higher velocity.
• Velocity vs. Time Graph:
  • Slope indicates acceleration.
  • Flat line indicates no acceleration (constant velocity).
• Interpretation:
  • Look for conditions of maximum acceleration (steepest slope) and constant velocity (horizontal line).

Relationships Between Variables

• Directly Proportional:
  • If velocity increases, distance also increases.
• Inversely Proportional:
  • If velocity increases, time decreases to cover the same distance.

Example Problems

1. Cheetah Acceleration Problem:
   • Given:
     • Initial Velocity ($V_i$) = 0 m/s,
     • Acceleration ($a$) = 6.4 m/s²,
     • Time ($t$) = 5 seconds.
   • Find Final Velocity ($V_f$) and Distance ($d$):
     • $V<em>f = a \cdot t$ \rightarrow $V</em>f = 6.4 \cdot 5 = 32 m/s$
     • $d = V_i t + \frac{1}{2} a t^2 \rightarrow d = 0 + \frac{1}{2}(6.4)(5^2) = 80 m$
2. Average Speed Problem:
   • Walk 360 m at 1.5 m/s, find time:
     • $t = \frac{d}{s} = \frac{360}{1.5} = 240 seconds = 4 minutes$

Projectile Motion

• Acceleration due to Gravity:
  • $g = 9.8 \, ext{m/s}^2$.
  • For vertical motion, initial conditions must take into account gravity acting as a downward acceleration (negative direction).
• Example:
  • An object thrown upwards to a height of 20 m:
    • Final velocity at maximum height ($V_f$) = 0.
    • Use kinematic formulas with $Vi$, $a = -9.8$, and $d = 20 m$ to find initial velocity ($Vi$).