CivilForge

Time-Mean and Space-Mean Speed

v_{t} = \frac{1}{n} \sum v_{i}, v_{s} = \frac{n}{\sum ( 1/ v _{i} )}

Time-mean speed vt is the arithmetic mean of spot speeds (what a radar gun averages). Space-mean speed vs is the harmonic mean of those speeds — equivalently total distance over total travel time — and is the v in q = k v. Always vs ≤ vt, with equality only if all vehicles travel at the same speed.

v_{t}

(mph)

Time-mean speed (arithmetic)

v_{s}

(mph)

Space-mean speed (harmonic)

v_{i}

(mph)

Spot speed of vehicle i

n

Number of observations

Step-by-step solved example

Three vehicles: 30, 40, and 60 mph. Find vt and vs.

Spot vs space-mean speed of the same vehicle stream.

1. Time-mean
vt = (30+40+60)/3 = 43.3 mph.
$v_{t} = 43.3 mph$
2. Space-mean
vs = 3 / (1/30 + 1/40 + 1/60) = 3 / (0.03333+0.025+0.01667) = 3/0.075 = 40.0 mph.
$v_{s} = 40.0 mph$

Answer: vt = 43.3 mph, vs = 40.0 mph

10 practice questions

0/10 correct

1.Two speeds 40 and 60 mph. vs =

2.For the same two speeds, vt =

3.The speed in q = k v must be

4.vs equals vt when

5.A trap of length 0.50 mi is timed at 40 s and 50 s for two cars. vs is nearest

6.Time-mean speed is what you get by averaging

7.Because slower vehicles occupy the road longer, space-mean is

8.Four identical 50 mph observations give vs =

9.vt = 60 mph, vs = 54 mph. The difference indicates

10.If travel time over 2 mi is 3 min, space-mean speed of that vehicle is