Principles of Motion: Assertions and Reasons in Kinematics

Distance and Displacement Relationship

• General Comparison (Assertion/Reason Context):
  • Reason (R): The magnitude of displacement is described as being always equal to or more than the distance travelled by a body.
  • Assertion (A): Distance travelled by a body is always positive.
  • Reason (R): Displacement of a body may be positive, negative, or zero.
  • Analysis of Sign Convention: While distance is a scalar quantity representing the total path length and is strictly non-negative ($\text{Distance} \ge 0$), displacement is a vector quantity representing the change in position. Since displacement depends on direction relative to an origin, it can take positive, negative, or zero values.

Kinematic Graph Limitations

• The Velocity-Time ($v-t$) Graph:
  • Assertion (A): A $v-t$ graph perpendicular to the time axis is not physically possible.
  • Reason (R): If a $v-t$ graph were perpendicular to the time axis, it would imply that the acceleration of the particle is infinite.
  • Mathematical Context: Acceleration ($a$) is defined as the time rate of change of velocity: $a = \frac{dv}{dt}$. A graph perpendicular to the time axis signifies that the change in time ($dt$) is zero for a finite change in velocity ($dv$). Thus, $a = \lim_{dt \to 0} \frac{dv}{dt} = \infty$. In physical systems, infinite acceleration is impossible as it would require infinite force.

Retardation and Velocity

• Directional Dynamics:
  • Assertion (A): Retardation (negative acceleration) is always directed opposite to the direction of the velocity.
  • Reason (R): Retardation is defined as being equal to the time rate of decrease of velocity.
  • Physical Meaning: If a body is moving in a positive direction with velocity ($v$) and its speed is decreasing, the acceleration ($a$) must be acting in the negative direction to oppose the motion.

Relative Velocity in Linear Paths

• Magnitude and Direction:
  • Assertion (A): The relative velocity of two particles moving on the same straight line path can be greater in magnitude than the velocity of either individual particle.
  • Reason (R): When two particles are moving with velocities $v_1$ and $v_2$ in opposite directions, their relative velocity ($v_{rel}$) is calculated as the sum of their magnitudes.
  • Formula: $v_{rel} = v_1 + v_2$. Because both magnitudes are added, the resulting relative velocity can exceed the individual values of $v_1$ or $v_2$.

Velocity, Speed, and Circular Motion

• Variable Velocity with Constant Speed:

  • Assertion (A): The velocity of a particle may vary even when its speed remains constant.
  • Reason (R): Such a body may be moving along a circular path.
  • Conceptual Distinction: Velocity is a vector (magnitude and direction), while speed is a scalar (magnitude only). In uniform circular motion, the speed ($|v|$) is constant, but the direction of motion changes continuously at every point on the path, meaning the velocity ($v$) is constantly varying.

• Acceleration in Circular Paths:

  • Assertion (A): A body moving on a circular path is accelerated.
  • Reason (R): Velocity changes due to the change in direction, even though the speed remains the same.
  • Specific Contextual Reason (Assertion 12): The motion of a body on a circular path is described in the text as being under gravity.

Position-Time ($p-t$) Graphs for Static Bodies

• Body at Rest:
  • Assertion (A): The position-time ($p-t$) graph for a body at rest is a straight line parallel to the time axis.
  • Reason (R): A body at rest does not change its position with the lapse of time.
  • Graphical Representation: If position ($x$) is plotted against time ($t$), and the body is at rest at position $x_0$, the equation of the graph is $x = x_0$. This results in a horizontal line with a slope of zero ($\frac{dx}{dt} = 0$), confirmed by the fact that velocity is zero for a body at rest.

Verification of Assertions and Reasons

• Provided Answers for Review:
  • 5: (b)
  • 6: (b)
  • 7: (a)
  • 8: (a)
  • 9: (b)
  • 10: (b)