Constant velocity motion Knowledge-Based Study

Study notes

1. What is Constant Velocity Motion?

   An object moves with constant velocity (also known as uniform motion) along a straight line when both its speed and its direction of motion remain unchanged over time. In one-dimensional kinematics, this means the object covers equal displacements in equal intervals of time, regardless of how small those time intervals are chosen.

   Key Physical Conditions

   • Zero Acceleration: Because velocity does not change with time ($\Delta v = 0$), the acceleration of the object is identically zero ($a = 0$).
   • Equivalence of Average and Instantaneous Velocity: At any instant, the instantaneous velocity is equal to the average velocity over any arbitrary time interval: $v = v_{\text{avg}}$.
   • Directional Constancy: In one dimension, motion occurs along a single axis (such as the $x$-axis). A positive velocity indicates motion in the positive direction, while a negative velocity indicates motion in the opposite direction.

   Mathematical Formulation

   Because velocity is constant, the relationship between position, velocity, and time is linear.

   The displacement $\Delta x$ over an elapsed time interval $\Delta t$ is given by:

   $$\Delta x = v \Delta t$$

   If the motion begins at an initial clock reading $t_0 = 0$ with an initial position $x_0$, the position $x$ at any subsequent time $t$ is expressed as:

   $$x(t) = x_0 + vt$$

   Understanding the Variables

   • $x(t)$: Position of the object at time $t$ (measured in meters, $\text{m}$).
   • $x_0$: Initial position of the object at $t = 0$ (measured in meters, $\text{m}$).
   • $v$: Constant velocity (measured in meters per second, $\text{m/s}$).
   • $t$: Elapsed time (measured in seconds, $\text{s}$).

   Notice that if $x_0 = 0$, the position is directly proportional to time: $x = vt$.

   Position vs. Time Graph ($x-t$)

   The position-time graph of an object moving with constant velocity is always a straight line.

   • Slope: The slope of the $x-t$ graph represents the velocity: $\text{Slope} = \frac{\Delta x}{\Delta t} = v$.
   • Positive Slope: The line rises to the right, indicating constant positive velocity.
   • Negative Slope: The line falls to the right, indicating constant negative velocity (motion toward decreasing coordinate values).
   • Zero Slope: A horizontal line indicates that position does not change ($v = 0$), meaning the object is stationary.
   • Vertical Intercept: The point where the line crosses the vertical axis corresponds to the initial position $x_0$.

   Velocity vs. Time Graph ($v-t$)

   Because velocity is invariant with respect to time, the $v-t$ graph is a horizontal line parallel to the time axis.

   • Slope: The slope of the $v-t$ graph is zero, confirming that acceleration $a = 0$.
   • Area Under the Curve: The geometric area between the velocity line and the time axis over an interval $\Delta t$ represents the displacement: $\text{Area} = v \times \Delta t = \Delta x$.

   Critical Distinctions

   • Constant Speed vs. Constant Velocity: An object can maintain a constant speed while changing direction (such as on a curved path), but in that case, its velocity is not constant. Constant velocity strictly requires both constant speed and a constant straight-line direction.
   • Position vs. Velocity: A zero position ($x = 0$) does not imply zero velocity; it merely indicates that the object is passing through the chosen origin. Conversely, an object at a large coordinate value can be at rest if its velocity is zero.
   • Sign of Velocity: In one-dimensional coordinates, the sign of $v$ indicates direction relative to the chosen reference frame, not whether the object is "slowing down" or "speeding up". A velocity of $-5\text{ m/s}$ represents the exact same speed as $+5\text{ m/s}$, directed in the opposite sense.