2.2 Speed and Velocity

Speed and velocity describe how objects move, but they capture different information. Speed tells you how fast something travels, while velocity tells you how fast and in what direction. Mastering the distinction between these two is essential for everything that follows in kinematics, especially acceleration and force.

Speed and Velocity

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Calculation of average speed and velocity

Average speed and average velocity both involve dividing by time, but they use different quantities in the numerator.

Average speed uses total distance (the entire path length traveled), making it a scalar quantity with magnitude only:

$\text{average speed} = \frac{\text{total distance}}{\text{total time}}$

For example, if a car travels 200 km in 2 hours, its average speed is 100 km/h. It doesn't matter whether the car drove in a straight line or took a winding route.

Average velocity uses displacement (the straight-line change from start to finish), making it a vector quantity with both magnitude and direction:

$\text{average velocity} = \frac{\text{displacement}}{\text{total time}}$

For example, if a plane flies 500 km east in 2 hours, its average velocity is 250 km/h east. The direction is part of the answer.

A critical consequence: if you drive somewhere and return to your starting point, your average velocity is zero (displacement is zero), but your average speed is not.

Common units include meters per second (m/s), kilometers per hour (km/h), and miles per hour (mph). In most physics problems, you'll want to work in m/s.

Speed vs velocity in scenarios

Speed only cares about how fast an object moves. It ignores starting and ending positions entirely. Velocity, on the other hand, tracks both how fast and in what direction the object's position is changing.

The distinction really shows up in two classic scenarios:

• Circular motion: A car driving around a circular track at a constant rate has constant speed, but its velocity is continuously changing because the direction of motion keeps changing. This is why uniform circular motion still involves acceleration.
• Oscillating motion: A pendulum swinging back and forth may pass through the same speed at symmetric points in its swing, but the velocity flips sign each time it reverses direction. At the endpoints of the swing, the instantaneous velocity is momentarily zero even though the pendulum is about to speed up again.

The takeaway: any time an object changes direction, its velocity changes even if its speed stays the same.

Instantaneous speed and velocity analysis

Average values describe motion over a time interval. Instantaneous values describe motion at a single moment.

Instantaneous speed is the magnitude of an object's speed at one specific instant. Think of it as what your speedometer reads right now:

$\text{instantaneous speed} = \lim_{\Delta t \to 0} \frac{\Delta \text{distance}}{\Delta t}$

Instantaneous velocity is the velocity at one specific instant, including direction. For a falling ball, you might say its instantaneous velocity is 9.8 m/s downward at $t = 1$ s:

$\text{instantaneous velocity} = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$

This limit definition is the foundation of calculus-based physics. In practice, here's how you use instantaneous values:

1. On a position-time graph, the instantaneous velocity at any point equals the slope of the tangent line at that point. A steeper tangent means a faster velocity.
2. If the tangent line's slope is changing, the object is accelerating or decelerating.
3. On a velocity-time graph, the instantaneous velocity is simply the value you read off the curve at that moment.

Being comfortable reading tangent lines on graphs is one of the most useful skills for this unit.

Motion and Trajectory

• Motion is the change in an object's position over time. Describing motion fully requires specifying both how far the object has moved and in what direction.
• Trajectory is the path an object follows through space. It depends on the object's initial velocity and any forces acting on it. In one dimension, the trajectory is simply a line, but the concept becomes more important when you reach projectile motion.
• Distance is the total path length traveled, a scalar that is always positive or zero.
• Displacement is the straight-line change in position from start to finish, a vector that can be positive, negative, or zero.
• Time is the fundamental parameter connecting all motion quantities. Every kinematic equation you'll encounter uses time to relate position, velocity, and acceleration.