MOTION IN A STRIGHT LINE

A. Position, Path Length, and Displacement

\[\text{Position} = \text{Location of an object along a straight path, denoted by coordinate } x.\]
\[\text{Path Length} = \text{Total distance traveled (scalar quantity)}.\]
\[\text{Displacement} (\Delta x) = x_2 - x_1\]

Properties & Key Points:

  • \( x \): Position of an object along a line.
  • \( x_1, x_2 \): Initial and final positions.
  • \(\Delta x\): Displacement.
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B. Average Velocity and Average Speed

\[\text{Average Velocity} (v_{\text{avg}}) = \frac{\Delta x}{\Delta t}\]
\[\text{Average Speed} = \frac{\text{Total Path Length}}{\Delta t}\]

Properties & Key Points:

  • \( \Delta x \): Displacement over time interval \(\Delta t\).
  • \(\Delta t\): Time interval.
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C. Instantaneous Velocity and Speed

\[\text{Instantaneous Velocity} (v) = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}\]
\[\text{Instantaneous Speed} = |v|\]
  • \( \frac{dx}{dt} \): Derivative of displacement with respect to time, representing velocity.
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D. Acceleration

\[\text{Acceleration} (a) = \frac{\Delta v}{\Delta t}\]

Variables:

Instantaneous Acceleration Formula
  • \( \Delta v \): Change in velocity over time \(\Delta t\).
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E. Equations of Motion for Constant Acceleration

\[v = u + at\]
\[s = ut + \frac{1}{2}at^2\]
\[v^2 = u^2 + 2as\]

Properties & Key Points:

  • \( u \): Initial velocity.
  • \( v \): Final velocity.
  • \( a \): Constant acceleration.
  • \( s \): Displacement.
  • \( t \): Time interval.
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F. Relative Velocity

\[\text{Relative Velocity of A with respect to B} = v_{\text{AB}} = v_{\text{A}} - v_{\text{B}}\]

Properties & Key Points:

  • \( v_{\text{A}} \): Velocity of object A.
  • \( v_{\text{B}} \): Velocity of object B.
  • \( v_{\text{AB}} \): Relative velocity of A with respect to B.
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G. Uniform and Non-Uniform Motion

Properties & Key Points:

  • Uniform Motion: Motion with constant speed along a straight line.
  • Non-Uniform Motion: Motion with changing speed or direction.
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H. Graphical Representation of Motion

Properties & Key Points:

  • Slope gives velocity.
  • Slope gives acceleration.
  • Area under the curve gives displacement.
  • Area under the curve gives change in velocity.
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I. Concept of Free Fall

\[v = u + gt \]
\[h = ut + \frac{1}{2}gt^2\]
\[v^2 = u^2 + 2gh\]

Properties & Key Points:

  • When an object falls under the influence of gravity alone (ignoring air resistance), it has an acceleration equal to \( g \approx 9.8 \, \text{m/s}^2 \).
  • \( g \): Acceleration due to gravity.
  • \( h \): Height or displacement during free fall.
  • \( u \): Initial velocity (often 0 if dropped).
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J. Important Concepts and Terms

Properties & Key Points:

  • Scalar Quantity: Has only magnitude (e.g., distance, speed).
  • Vector Quantity: Has both magnitude and direction (e.g., displacement, velocity, acceleration).
  • Uniform Acceleration: Acceleration remains constant over time.
  • Instantaneous Velocity: The velocity of an object at a specific point in time.
  • Distance vs. Displacement: Distance is total length traveled (scalar), whereas displacement is the straight-line distance between start and end points (vector).
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