What is differentiation in a single variable?
Earth and Atmospheric Sciences
In common conversation, we use “speed” and “velocity” interchangeably, but in physics, they are distinct. Velocity is the rate at which an object changes its position.
The key difference? Direction. —
1. Speed vs. Velocity
- Speed is a scalar quantity. It only measures “how fast” an object is moving (e.g., 60 mph).
- Velocity is a vector quantity. It measures “how fast” and “in what direction” (e.g., 60 mph North).
If you drive in a perfect circle at a constant 60 mph, your speed is constant, but your velocity is constantly changing because your direction is changing every second.
2. The Mathematical Definition
In its simplest form (average velocity), the formula is:
vavg = Δs / Δt
Where:
- Δs is the displacement (the straight-line distance from start to finish).
- Δt is the time taken.
3. Instantaneous Velocity (The Calculus Connection)
This ties back to the Tangent Problem we discussed. If you want to know your velocity at one exact moment in time—rather than an average over a whole trip—you use a derivative.
If s(t) is a function representing your position at time t, then your velocity v(t) is the derivative of that position:
v(t) = ds / dt
On a graph of position vs. time, the velocity at any point is the slope of the tangent line.
4. Key Characteristics
- Units: Measured in meters per second (m/s), miles per hour (mph), or kilometers per hour (km/h).
- Positive vs. Negative: Because direction matters, velocity can be negative. If “North” is positive, traveling “South” gives you a negative velocity.+1
- Relationship to Acceleration: Acceleration is the rate at which velocity changes. If you change your speed OR your direction, you are accelerating.