B. Kinematics · Kinematics: Motion in One and Two DimensionsSpeed and Velocity: Average, Constant and Instantaneous
B2.1 B3.1 B3.2 B2.7About 70 minutes
Learning goals- I can calculate average speed and average velocity and explain how they differ.
- I can distinguish average, instantaneous and constant (uniform) velocity, with examples.
- I can rearrange v = Δd ÷ Δt to find displacement or time.
- I can convert between km/h and m/s.
Words to know
average speed (vₐᵥ)
Total distance travelled ÷ total time. A scalar.
average velocity (v⃗ₐᵥ)
Displacement ÷ time interval: vₐᵥ = Δd ÷ Δt. A vector in the direction of the displacement.
instantaneous velocity
The velocity at one particular instant, such as the reading on a speedometer plus the direction of travel.
uniform (constant) velocity
Motion with the same speed in the same direction, so equal displacements happen in equal time intervals.
non-uniform motion
Motion in which the speed, the direction, or both change.
m/s
The SI unit of speed and velocity: metres per second. 1 m/s = 3.6 km/h.
Two very different "fast"s
A long-distance runner finishes a 400 m lap of the school track in 80 s. She was clearly moving quickly the whole time. Yet she finished exactly where she started. Was she fast or not?
Both, depending on what you measure. Her speed was high, but her average velocity for the lap was zero. Just as distance and displacement are different, speed and velocity are different, and you need both.
Average speed and average velocity
Average speed uses distance, so it is a scalar:
average speed = total distance ÷ total time
Average velocity uses displacement, so it is a vector:
vₐᵥ = Δd ÷ Δt
The direction of the average velocity is the direction of the displacement. The SI unit for both is metres per second (m/s). For road travel we often use kilometres per hour (km/h).
If an object moves in one direction without turning back, average speed equals the magnitude of average velocity. If it turns back or changes direction, average speed is larger.
Worked exampleExample 1: One lap of the track
A runner completes one 400 m lap in 80.0 s and finishes at the start line. Find her average speed and average velocity.
- Average speed = 400 m ÷ 80.0 s = 5.00 m/s.
- She ends where she started, so Δd = 0 m.
- vₐᵥ = Δd ÷ Δt = 0 m ÷ 80.0 s = 0 m/s.
Average speed = 5.00 m/s; average velocity = 0 m/s.
Worked exampleExample 2: A road trip with a stop
A family drives 150 km [E] in 1.5 h, stops for lunch for 0.50 h, then realizes they forgot the dog's leash and drives 50 km [W] in 0.50 h. Find (a) the average speed and (b) the average velocity for the whole trip.
- Total time includes the stop: Δt = 1.5 + 0.50 + 0.50 = 2.5 h.
- (a) Total distance = 150 + 50 = 200 km. Average speed = 200 km ÷ 2.5 h = 80 km/h.
- (b) East positive: Δd = +150 + (−50) = +100 km = 100 km [E].
- vₐᵥ = 100 km [E] ÷ 2.5 h = 40 km/h [E].
- The average speed is twice the average velocity's magnitude because part of the trip went backward.
(a) 80 km/h; (b) 40 km/h [E].
Average, instantaneous and constant
- Average velocity describes a whole time interval. It does not tell you what happened moment by moment.
- Instantaneous velocity is the velocity at one instant. A car's speedometer shows instantaneous speed; add the direction of travel and you have instantaneous velocity.
- Constant (uniform) velocity means the instantaneous velocity never changes: same speed, same direction. Then the object moves equal displacements in equal time intervals, and its average velocity over any interval equals its instantaneous velocity.
Motion that is not uniform is called non-uniform: the object speeds up, slows down or changes direction. Most real motion is non-uniform. A car driving through a city has an average velocity for the trip, but its instantaneous velocity changes at every light.
Rearranging the equation
The equation vₐᵥ = Δd ÷ Δt has three variables, so if you know any two you can find the third:
Make sure the units match before you calculate. If the speed is in m/s, time must be in seconds and distance in metres. If the speed is in km/h, use hours and kilometres.
Worked exampleExample 3: Finding displacement and time
(a) A GO train travels at a constant velocity of 25 m/s [W] for 3.0 min. Find its displacement. (b) A student walks to school, 1.2 km away, at an average speed of 1.5 m/s. How many minutes does the walk take?
- (a) Convert time: Δt = 3.0 min × 60 s/min = 180 s.
- Δd = vₐᵥ Δt = (25 m/s [W])(180 s) = 4500 m [W] = 4.5 km [W].
- (b) Convert distance: 1.2 km = 1200 m.
- Δt = d ÷ v = 1200 m ÷ 1.5 m/s = 800 s.
- 800 s ÷ 60 s/min ≈ 13 min.
(a) 4.5 km [W]; (b) about 13 min.
Worked exampleExample 4: Converting units
The speed limit on many Ontario highways is 100 km/h. (a) Express this in m/s. (b) A hockey slapshot travels at 45 m/s. Express this in km/h.
- (a) 100 km/h ÷ 3.6 = 27.77... m/s ≈ 27.8 m/s.
- (b) 45 m/s × 3.6 = 162 km/h (or 1.6 × 10² km/h to 2 significant digits).
(a) 27.8 m/s; (b) about 1.6 × 10² km/h.
Just for funI ran the fastest lap of my life today: 400 m in 70 s. My coach checked the math and told me my average velocity was zero.
I've never worked so hard to go nowhere.
Hands-on activityHow fast do you walk?
You need: a measuring tape (or a known distance such as sidewalk squares you have measured), a stopwatch or phone timer, and a partner. Work on a quiet sidewalk, path or driveway, away from traffic.
- Mark a straight 20.0 m course with chalk or two small objects.
- Time yourself walking the course at a normal pace. Repeat twice and average the times.
- Time a fast walk the same way.
- Now walk 10.0 m out, turn around, and walk back to the start. Time the whole trip.
- For each walk, calculate the average speed in m/s and km/h. For step 4, also calculate the average velocity.
What to look for: a typical normal walking speed is around 1.2 to 1.5 m/s (about 4 to 5 km/h). In step 4 the average speed is about the same as your normal walk, but the average velocity is 0 m/s because the displacement is zero. Times should be recorded to 0.1 s; human reaction time makes more precision meaningless.
Your turn
1A cyclist rides 6.0 km in 0.25 h. What is her average speed in km/h and in m/s?
6.0 ÷ 0.25 = 24 km/h. 24 ÷ 3.6 = 6.7 m/s.
2A ball rolls 3.0 m [E] in 2.0 s and then 1.0 m [W] in 2.0 s. Find its average speed and average velocity for the 4.0 s.
Distance = 4.0 m; average speed = 4.0 ÷ 4.0 = 1.0 m/s. Δd = 3.0 − 1.0 = 2.0 m [E]; vₐᵥ = 2.0 ÷ 4.0 = 0.50 m/s [E].
4How far does a car travelling at 30.0 m/s [N] go in 12.0 s?
Δd = 30.0 × 12.0 = 360 m [N].
5A canoe paddles 2.4 km at an average speed of 1.6 m/s. How long does it take, in minutes?
2400 m ÷ 1.6 m/s = 1500 s = 25 min.
6Which reading tells you instantaneous speed: a speedometer, or a trip computer showing "average 68 km/h"? Explain.
The speedometer. It shows how fast the car is moving at that moment. The trip computer divides total distance by total time, which is an average.
More practice: warm-up
1Write the equation for average velocity.
2What is the SI unit for velocity?
4A car moves at a steady 60 km/h [S]. Is its velocity uniform? Explain.
Yes, as long as both the speed (60 km/h) and the direction [S] stay the same.
5A snail crawls 12 cm in 60 s. What is its average speed in cm/s?
More practice: on your own
1A plane flies 840 km [W] in 1.5 h. Find its average velocity in km/h and m/s.
840 ÷ 1.5 = 560 km/h [W]; 560 ÷ 3.6 ≈ 156 m/s [W] (1.6 × 10² m/s to 2 significant digits).
2A dog runs 25 m [E] in 5.0 s, rests 3.0 s, then runs 15 m [W] in 2.0 s. Find its average speed and average velocity.
Distance 40 m, time 10.0 s: average speed = 4.0 m/s. Δd = 25 − 15 = 10 m [E]; vₐᵥ = 10 ÷ 10.0 = 1.0 m/s [E]. The rest time counts in Δt.
3Light travels at 3.00 × 10⁸ m/s. Sunlight takes about 499 s to reach Earth. Estimate the Earth–Sun distance.
d = vΔt = (3.00 × 10⁸)(499) ≈ 1.50 × 10¹¹ m (150 million km).
4A student's average speed for a trip is 5.0 m/s, but her average velocity is 3.0 m/s [N]. What does this tell you about her path?
Her distance was larger than the magnitude of her displacement, so she did not travel in one straight direction: she changed direction or turned back for part of the trip.
5Explain why all three objects in the motion diagram above have the same average speed even though they move differently.
Each one covers the same total distance (5 spacing units) in the same 5.0 s. Average speed only depends on total distance and total time, not on how the speed changed along the way.
Challenge
1You drive 60 km at 60 km/h and then 60 km at 120 km/h. What is your average speed for the trip? (Hint: it is not 90 km/h.)
Times: 60/60 = 1.0 h and 60/120 = 0.50 h. Average speed = 120 km ÷ 1.5 h = 80 km/h. You spend more time at the slower speed, so the average is pulled toward 60 km/h.
2Two friends start 3.0 km apart on a straight trail and walk toward each other, one at 1.4 m/s and the other at 1.1 m/s. How long until they meet, and how far does the faster one walk?
They close the gap at 1.4 + 1.1 = 2.5 m/s. Δt = 3000 ÷ 2.5 = 1200 s (20 min). The faster friend walks 1.4 × 1200 = 1680 m ≈ 1.7 km.
3A car's average velocity from 1:00 p.m. to 3:00 p.m. was 50 km/h [E]. Can you be sure the speedometer read 50 km/h at 2:00 p.m.? Explain.
No. Average velocity describes the whole interval. The car might have been stopped, going faster, or even going west at 2:00 p.m. Only if the velocity was constant would the instantaneous value equal the average.
Quiz
1. Average velocity is defined as:
3. A hiker walks 4.0 km [N] in 1.0 h and then 4.0 km [S] in 1.0 h. Her average velocity is:
4. Which situation shows uniform motion?
5. A car travels at 20 m/s [E] for 45 s. Its displacement is:
Big idea: Speed (scalar) and velocity (vector) are different, and average, instantaneous and constant values mean different things (B3.1, B3.2). Students also begin solving one-dimensional motion problems algebraically (B2.7).
How to help: On a car or bus trip, compare the speedometer (instantaneous) with total distance ÷ total time (average). Ask: "Would our average velocity be different from our average speed if we came home the same way?"
Common difficulties:
- Using distance where displacement is needed (or the reverse).
- Leaving out stopped time when calculating an average.
- Mixing units: km/h with seconds, or km with m/s. Have your teen write units in every line.
- Averaging two speeds directly (Challenge question 1). The average must be total distance ÷ total time.