2bSpeed and Velocity
Distinguish average speed from average velocity, calculate instantaneous values, and understand why direction transforms speed into velocity.
Speed and velocity appear in every branch of physics and engineering. Whether designing a safe highway, calculating a satellite's orbit, or analyzing a collision, you must know whether direction matters — and it almost always does.
Lesson Overview
How fast is an object moving, and in which direction? In this lesson you will distinguish between speed (a scalar) and velocity (a vector), calculate average and instantaneous values of each, and apply these concepts to real-world motion problems. These ideas are the core of kinematics.
Key Concepts
Average Speed
Total distance divided by total time: s_avg = d / t
Average Velocity
Displacement divided by time interval: v_avg = Δx / Δt
Instantaneous Speed
The speed at a specific instant — the magnitude of instantaneous velocity
Instantaneous Velocity
The velocity at a specific instant — the limit of Δx/Δt as Δt → 0
Speed (scalar)
Magnitude of motion — always ≥ 0, no direction
Velocity (vector)
Rate of change of position — has magnitude and direction
A car travels 150 km in 2.5 hours. Calculate its average speed.
A runner starts at x = 0, runs to x = 100 m, then returns to x = 40 m. The total time is 20 s. Find (a) average speed and (b) average velocity.
A car completes a 400 km round trip in 5 hours. What is (a) the average speed and (b) the average velocity?
A cyclist travels at 12 m/s east for 10 s, then at 8 m/s west for 5 s. Find the average velocity for the entire trip.
A speedometer reads 60 km/h. Is this a speed or a velocity? Explain.
External Supplemental Resource
Guided Practice Video: Speed and Velocity
Review average speed and average velocity with worked examples before completing the guided practice problems below.
Video by Khan Academy on YouTube
Watch on YouTube ↗A train travels 300 km north in 3 hours. Find its average velocity.
Hint: Average velocity = displacement / time. Include direction in your answer.
A swimmer swims 4 laps in a 25 m pool in 80 s. Find (a) average speed and (b) average velocity.
Hint: 4 laps = 4 × 25 m of distance. What is the displacement after 4 laps (even number of laps)?
Can an object have a constant speed but changing velocity? Give an example.
Hint: Think about circular motion.
A car's position is given by x = 5t² (in metres, t in seconds). What is the instantaneous velocity at t = 3 s?
Hint: Instantaneous velocity = dx/dt. Differentiate x with respect to t, then substitute t = 3.
Two cars travel between the same two cities. Car A takes 2 h; Car B takes 3 h. Which has greater average speed? Which has greater average velocity?
Hint: Both travel the same displacement. Compare d/t for each.
Key Vocabulary
Average Speed
The total distance traveled divided by the total time taken. A scalar quantity.
Example: Driving 120 km in 2 hours gives an average speed of 60 km/h.
Average Velocity
The total displacement divided by the total time interval. A vector quantity.
Example: If you end up 60 km north of your start after 2 hours, your average velocity is 30 km/h north.
Instantaneous Velocity
The velocity of an object at a single moment in time; the derivative of position with respect to time.
Example: A car's speedometer shows instantaneous speed; adding direction gives instantaneous velocity.
Uniform Motion
Motion at constant velocity — constant speed in a straight line.
Example: A car on a straight highway at a steady 100 km/h is in uniform motion.
Interactive Practice — 5 Questions
Average velocity is defined as:
A car travels 100 km east then 100 km west in 4 hours. What is the average velocity?
Which of the following is a scalar quantity?
A runner completes 400 m in 50 s. What is the average speed?
Instantaneous velocity is:
Independent Practice
A jogger runs 2 km east in 12 min, then 1 km west in 8 min. Find (a) average speed and (b) average velocity for the entire run.
Explain why average speed can never be negative, but average velocity can be.
A car travels at 30 m/s for 4 s, then at 20 m/s for 6 s in the same direction. Calculate the average speed for the entire trip.
A ball is thrown straight up at 15 m/s. At the top of its path, what is the instantaneous speed? What is the instantaneous velocity?
★ A particle's position is given by x(t) = 3t³ − 12t (in metres, t in seconds). Find the instantaneous velocity at t = 2 s and determine whether the particle is moving in the positive or negative direction at that moment.
ChallengeCommon Mistakes
Using distance instead of displacement when calculating average velocity.
Average velocity = displacement / time. Average speed = distance / time. These give different answers when direction changes.
Assuming average speed equals the average of two speeds.
Average speed = total distance / total time. The average of two speeds only works if equal distances (not times) are spent at each speed.
Saying velocity is always positive.
Velocity is a vector — it is negative when motion is in the negative direction (e.g., moving left when right is positive).
Math Tips
Key formulas: Average speed = d_total / t_total. Average velocity = Δx / Δt = (x_f − x_i) / (t_f − t_i). Always check: did you use distance or displacement?
Instantaneous velocity = dx/dt (the derivative of position). If x = at² + bt + c, then v = 2at + b.