Velocity: Speed with Direction

Now that we can describe position, let's describe how position changes over time. This brings us to velocity - one of the most important concepts in physics.

Position-Time Graph

The slope of a position-time graph gives velocity

Illustration: velocity-graph

Average Velocity

v_avg = Δx / Δt = (x_f - x_i) / (t_f - t_i)

The rate of change of position. How fast position changes, and in what direction.

v_avg:: Average velocity (m/s)
Δx:: Displacement (m)
Δt:: Time interval (s)

⚠️

Speed vs Velocity

Speed is how fast you're going (always positive).
Velocity is how fast AND in what direction (can be positive or negative).

A car going 60 mph north has velocity +60 mph.
A car going 60 mph south has velocity -60 mph (if north is positive).
Both have the same speed: 60 mph.

Instantaneous Velocity

v = lim[Δt->0] Δx/Δt = dx/dt

The velocity at a specific instant in time. This is the derivative of position with respect to time.

Average vs Instantaneous

Easy

A car travels 150 km in 2 hours. For the first hour it went 60 km/h, for the second hour it went 90 km/h. Find the average velocity.

1

Use the average velocity formula

v_avg = Δx / Δt = 150 km / 2 hr = 75 km/h

2

Note the difference

The instantaneous velocities were 60 and 90 km/h, but the average was 75 km/h.

Answer: 75 km/h

Practice

1. A runner completes a 400m track in 50 seconds, ending where she started. What is her average velocity?

8 m/s

0 m/s

400 m/s

50 m/s

Average velocity = displacement / time. Since she ended where she started, displacement = 0, so average velocity = 0. Her average SPEED was 8 m/s, but velocity accounts for direction.

Key Takeaways

Velocity = displacement / time (vector)
Speed = distance / time (scalar, always positive)
Average velocity looks at the overall change
Instantaneous velocity is velocity at one moment (derivative of position)