Velocity

The mathematical relationship for average velocity is the ratio of total displacement to the total time interval: $v_{avg} = \frac{\Delta s}{\Delta t}$ where $\Delta s$ is the change in position and $\Delta t$ is the time elapsed.

In calculus-based physics, instantaneous velocity is the first derivative of the position function with respect to time: $v(t) = \frac{ds}{dt}$

This derivative represents the slope of the tangent line on a position-time ($s-t$) graph at any specific moment.

On a Velocity-Time (v-t) graph, the vertical axis represents velocity and the horizontal axis represents time. The behavior of the line provides information about the object's motion.

Gradient (Slope): The slope of a $v-t$ graph represents the acceleration. A positive slope indicates increasing velocity, a negative slope indicates deceleration, and a horizontal line indicates constant velocity.

Area Under the Curve: The geometric area between the velocity line and the time axis represents the displacement (or distance travelled if the velocity is always positive).

For constant velocity, the area is a rectangle ($v \times t$). For constant acceleration starting from rest, the area is a triangle ($\frac{1}{2} \times v \times t$).

Check the Units: Always ensure time is in seconds and displacement is in metres before calculating. If given $km/h$, divide by $3.6$ to convert to $m/s$.

Direction Matters: In one-dimensional motion, use signs ($+$ or $-$) to indicate direction. Moving 'backwards' results in a negative velocity.

Graph Interpretation: If a $v-t$ graph goes below the x-axis, the object has changed direction. The area below the axis represents negative displacement.

Sanity Check: If an object returns to its start point, its average velocity for the whole trip MUST be zero, regardless of how fast it moved during the journey.

Distance vs. Displacement: A common error is using the total path length (distance) to calculate velocity. Velocity requires the straight-line change in position (displacement).

Constant Speed vs. Constant Velocity: An object moving in a circle at a steady $10 m/s$ has constant speed but changing velocity because its direction is constantly shifting.

Zero Velocity vs. Zero Acceleration: An object can have zero velocity for an instant (like a ball at the peak of its throw) while still having non-zero acceleration (gravity).