Position, Distance, and Displacement in One-Dimensional Motion

Distinction Between Instant of Time and Time Interval

• Instant of Time:

  • Defined as a single reading of a clock at a given point of time.
  • It represents a specific moment rather than a duration.

• Time Interval:

  • Defined as the time duration between two instants of time.
  • It is calculated as the difference between two readings of a clock.

Describing Position and Direction

• Reference Point:

  • To describe the position of an object, a starting point must be established as a reference point.
  • In the provided example of an athlete named Neena, the starting point is marked as the origin $O$.

• The Coordinate System:

  • Positions are marked on a straight line with distances relative to the origin.
  • As shown in Fig. 4.3, the origin is $0\,m$, and positions such as point $B$ ($40\,m$) and point $A$ ($100\,m$) are marked along the track.

• Specifying Direction:

  • For an object moving in a straight line, there are only two possible directions: forward and backward.
  • Direction is represented using plus ($+$) and minus ($-$) signs.
  • Convention: Positions to the right of the reference point $O$ are generally taken as positive ($+$), and positions to the left of $O$ are taken as negative ($-$).

Scalars and Vectors

• Scalars:

  • Physical quantities which can be specified by just their numerical value are called scalars.

• Vectors:

  • Physical quantities which require specifying both the direction and magnitude are called vectors.

• Magnitude:

  • The numerical value (with units) of a physical quantity is called its magnitude.

Distance Travelled and Displacement

• Distance Travelled:

  • Refers to the total path length covered by an object between its starting and stopping positions.
  • It is a scalar quantity as it does not require a specified direction.

• Displacement:

  • Defined as the net change in the position of an object between two given instants of time.
  • A complete description of displacement requires specifying both a direction and its numerical value (magnitude) with units.
  • Direction of Displacement: Specified from the position at the first instant toward the position at the second instant.
  • Magnitude of Displacement: Represents the straight-line distance between the object's positions at the two specific instants.

Comprehensive Example: Motion of an Athlete on a Straight Track

Consider an athlete running on a straight track (as illustrated in Fig. 4.4):

• Timeline of Motion:

  • At time $t = 0\,s$: The athlete starts at point $O$ ($0\,m$).
  • At time $t = 4\,s$: The athlete reaches point $B$ ($40\,m$).
  • At time $t = 10\,s$: The athlete reaches point $A$ ($100\,m$).
  • At time $t = 16\,s$: The athlete runs back along the same path and reaches point $B$ again ($40\,m$).

• Calculation of Total Distance Travelled:

  • The athlete first moves from $O$ to $A$, then from $A$ back to $B$.
  • $\text{Total distance} = OA + AB$
  • $\text{Total distance} = 100\,m + 60\,m = 160\,m$

• Calculation of Displacement:

  • Displacement is the change in position from the start ($t = 0\,s$ at point $O$) to the finish ($t = 16\,s$ at point $B$).
  • $\text{Displacement} = OB = 40\,m$
  • Note: The displacement ($40\,m$) is different from the total distance travelled ($160\,m$).