Speed and velocity Knowledge-Based Study

Study notes

1. Describing How Fast an Object Moves

   In kinematics, describing motion requires more than just knowing how far an object travels; we also need to know how rapidly that change of position occurs over time.

   Before distinguishing between speed and velocity, recall two fundamental concepts of distance and position:

   • Distance ($d$): The total path length traversed by an object during its motion. Distance is a scalar quantity, meaning it has magnitude but no direction.
   • Displacement ($\Delta x$): The straight-line change in position from an initial point to a final point ($\Delta x = x_f - x_i$). Displacement is a vector quantity, possessing both magnitude and a specific spatial direction.

   Because motion can be quantified using either total path length or net change in position, physics defines two distinct rates of motion: speed and velocity.

   Defining Speed

   Speed is the rate at which an object covers distance. Because it depends directly on distance, speed is a scalar quantity; it is completely described by a numerical value and a unit, with no reference to direction.

   Average Speed

   The average speed ($v_{\text{avg}}$) of an object over a given time interval is defined as the total distance traveled divided by the total elapsed time:

   $$v_{\text{avg}} = \frac{d}{\Delta t}$$

   • $d$: Total distance traveled (measured in meters, $\text{m}$)
   • $\Delta t$: Total elapsed time (measured in seconds, $\text{s}$)
   • SI Unit: Meters per second ($\text{m/s}$)

   Instantaneous Speed

   While average speed describes the overall rate of travel across an entire journey, instantaneous speed is the speed of an object at a specific, exact moment in time. A vehicle's speedometer displays instantaneous speed.

   Note: Because distance can never be negative, speed is always a non-negative quantity ($v \ge 0$).

   Defining Velocity

   Velocity is the rate of change of position with respect to time. Because it depends on displacement, velocity is a vector quantity; it requires both a magnitude (how fast the object moves) and a direction (where it is heading).

   Average Velocity

   The average velocity ($\vec{v}_{\text{avg}}$) of an object over a time interval is defined as the net displacement divided by the time interval:

   $$\vec{v}_{\text{avg}} = \frac{\Delta \vec{x}}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$$

   • $\Delta \vec{x}$: Displacement vector (meters, $\text{m}$)
   • $\Delta t$: Elapsed time (seconds, $\text{s}$)
   • SI Unit: Meters per second ($\text{m/s}$), accompanied by a directional indicator (such as north, east, or a positive/negative sign in one-dimensional motion).

   Constant vs. Changing Velocity

   An object moves with constant velocity only if both its speed and its direction of motion remain completely unchanged. If an object turns, even while maintaining a constant numerical speed, its velocity changes because its direction of motion has changed.

   Key Differences and Practical Analysis

   Understanding the distinction between speed and velocity is crucial when evaluating round trips and curved paths.

   Comparison Summary

   • Nature: Speed is scalar; velocity is vector.
   • Sign: In one dimension, speed is always positive or zero, whereas velocity can be positive or negative depending on the chosen coordinate axis.
   • Magnitude Relation: The magnitude of average velocity is not always equal to average speed, although instantaneous speed is always equal to the magnitude of instantaneous velocity.

   Illustrative Scenario: A Closed Loop

   Consider an athlete who runs one complete lap around a 400-meter circular track in 50 seconds, returning to the exact starting point:

   • Total distance ($d$): $400\text{ m}$
   • Net displacement ($\Delta x$): $0\text{ m}$ (since initial and final positions are identical)
   • Average speed: $$v_{\text{avg}} = \frac{400\text{ m}}{50\text{ s}} = 8\text{ m/s}$$
   • Average velocity: $$\vec{v}_{\text{avg}} = \frac{0\text{ m}}{50\text{ s}} = 0\text{ m/s}$$

   This classic example demonstrates that an object can maintain a high average speed throughout an interval while having an average velocity of zero.