2.3 Time, Velocity, and Speed

Velocity and speed describe how objects move through space and time. These concepts are the foundation of kinematics, and you'll rely on them constantly as you move into forces, energy, and more complex motion problems.

The key skill here is understanding the difference between instantaneous and average velocity, and knowing how to extract motion information from graphs. Once these click, you'll be able to calculate displacements, interpret position-time and velocity-time graphs, and predict where an object will be at a given time.

Velocity and Speed

Instantaneous vs. average velocity

Velocity is a vector quantity that describes the rate of change of an object's position in a particular direction. Because it's a vector, velocity has both magnitude and direction.

There are two types you need to know:

• Instantaneous velocity is the velocity at one specific moment in time. Think of it as what your speedometer reads right now (plus a direction). On a position vs. time graph, it equals the slope of the tangent line at that point.
• Average velocity is the total displacement divided by the total time interval:

$v_{avg} = \frac{\Delta x}{\Delta t}$

where $\Delta x$ is the change in position and $\Delta t$ is the change in time. On a position vs. time graph, this equals the slope of the secant line connecting two points.

Displacement ($\Delta x$) is the change in position from start to finish, including direction. It's a vector quantity. This is different from distance, which is the total path length traveled regardless of direction.

For example, if you walk 3 m east and then 1 m west, your displacement is 2 m east, but your total distance traveled is 4 m.

Velocity and speed calculations

To calculate velocity:

$v = \frac{\Delta x}{\Delta t}$

Velocity is positive when the object moves in the positive direction and negative when it moves in the negative direction. By convention, rightward motion is typically positive and leftward is negative.

Speed is the magnitude of velocity, with no direction attached. It's a scalar quantity. To calculate average speed:

$s = \frac{d}{\Delta t}$

where $d$ is the total distance traveled (not displacement). Speed is always positive or zero.

Here's where students often slip up: average speed and the magnitude of average velocity are not always the same. Using the walking example above, if the whole trip took 4 seconds, your average speed is $\frac{4 \text{ m}}{4 \text{ s}} = 1 \text{ m/s}$, but your average velocity magnitude is $\frac{2 \text{ m}}{4 \text{ s}} = 0.5 \text{ m/s}$.

Position-time graphs for velocity

The slope of a position vs. time graph gives you the velocity of the object.

• A straight line means constant velocity. The steeper the line, the faster the object moves. A line at 45° on a graph with matching scales corresponds to 1 m/s.
• A curved line means the velocity is changing. To find the instantaneous velocity at any point on a curve, draw the tangent line at that point and calculate its slope.
• A horizontal line means zero velocity, so the object is at rest.
• A positive slope (line going upward to the right) means motion in the positive direction. A negative slope (line going downward to the right) means motion in the negative direction.

Interpretation of velocity-time graphs

Velocity-time graphs encode two important pieces of information:

Area under the curve = displacement. Areas above the time axis count as positive displacement, and areas below count as negative displacement. The total displacement is the sum of these signed areas. You can often calculate these areas using simple geometry (rectangles, triangles, trapezoids).

Slope of the curve = acceleration. This is a preview of what you'll study next in kinematics.

• A straight line on a velocity-time graph means constant acceleration, with the slope equal to that acceleration.
• A curved line means the acceleration itself is changing. The instantaneous acceleration at any point equals the slope of the tangent line there.
• A horizontal line means the velocity isn't changing, so the acceleration is zero.

Motion and Reference

• Kinematics is the branch of physics that describes motion without considering the forces that cause it. You're learning to describe what happens before asking why.
• Frame of reference is the coordinate system from which you observe and measure motion. Your choice of reference frame determines the signs and values of position, velocity, and displacement.
• Relative motion describes how an object's motion appears differently depending on which frame of reference you use. A passenger sitting still on a moving train has zero velocity relative to the train but nonzero velocity relative to the ground.
• Newton's laws of motion, which you'll encounter soon, connect forces to the motion concepts you're building here.