Accid. Anal & Prey. Vol. fi, pp. 357-359. Pergamon Press 1973. Printed in Great Britain.

RESEARCH NOTE A R E M A R K O N D R I V E R ESTIMATION OF THE TIME OF

COLLISION WITH AN ONCOMING CAR

ALAN SOLOMON

Department of Mathematics, University of the Negev, Beersheva, Israel

(Receiced 13 June 1972)

I N T R O D U C T I O N

IT IS known that drivers tend to underestimate low speeds o f oncoming cars while over- estimating high speeds (see Salvatore, 1967; G o r d o n and Mast, 1968). For the purpose o f modeling driver behavior when passing, or of designing an automated passing procedure, it is convenient and natural to estimate the durat ion of time available before a potential collision, by the value

initial distance between vehicles time ---= (I)

closing speed

In this note we show that under certain conditions the observed interval of time given by equation (1) will tend to exceed the actual time.

STATEMENT OF THE PROBLEM

Given two vehicles I, I I moving towards each other with constant speeds ~t,Oz, along a straight road. At time t = 0 the driver o f I observes that I I is located ahead of him, at a distance do. It is known (Farber and Silver, t967) that do can be estimated accurately under most conditions, and we will assume that the estimate is exact. In addit ion the driver estimates the speed ~z of II as the value of a r andom variable v2. Assume that he estimates the time tes, at which I ,II collide if he should continue moving with the same speed and in the same direction as

do ros, - _ (2)

Ut ~ U2

while the actual time t~o. o f collision is

do tcon - - _ . (3)

vl + ~z

What is the relationship between these two values ? The estimated speed v2 is a r andom variable with mean value E(vz). We will now show

that if

e(v,_) ~ e2 (4) 357

358 ALAN SOLOMON

then the expected value of the random variable te~, obeys the condition

E(Gst) > t¢ou (5)

and hence, that the driver will tend to overestimate the collision time to a possibly dangerous extent.

We will now derive equation (5). By the basic properties of the expected value

E(t.~t) -- tcou = E ( t o s t - - t¢ott)

but by equations (2) and (3),

tes t - - tco[l = do(9, - v,)

(91 + 9~)(~ + v~)"

Using the relation

I 0 = 1

i + 0 1 + 0

with

(6)

we see that

whence

and so

v2 - - 92 0 - -

91 + g2

~1 + v2 ( ~ + ~2) i -- vl +

do { (~_, - ~_,)2~ t.,, --t~ou - - ( g t + 92)_. (92 - -v2 ) + - (7)

_ do E(92 - - v 2 ) + E ( 8 ) E(t,,t - - t¢ou) (gt -~ u2) 2 (ul -~ 02) 2 k Ul @ 92 J

But by equation (4), the first term on the right is non-negative, and the second term is positive unless v2 is always equal to g,_, i.e. unless the driver always estimates the speed of the opposing vehicle exactly.

IMPLICATIONS OF EQUATION (5)

The relation (5) has several clear implications. The first is that when a driver consistently does not overestimate v2, he must compensate for equation (5) in passing by underestimating the collision time given by equation (3), in order to avoid a collision. The second is that in building a mathematical simulation for passing, the collision time should be weighted in accordance with equation (8), on the assumption that a driver instinctively does this. The third is that inherent dangers would exist if an automated passing procedure were to be effected without the further examination of all effects of errors in the system.

Driver estimation of the time of collision with an oncoming car 359

R E F E R E N C E S

FARBER, E. and SILVER, C. (1967) Knowledge of oncoming car speed as a determiner of driver's passing behavior. Highway Res. Rec. 195, 52-65.

GORDOy, D. and MAST, T. (1968) Drivers decisions in overtaking and passing. Highway Res. Rec. 247, 42-50.

SALV,',XORE, S. (1967) Estimation of vehicular velocity under time limitation and restricted conditions of observation. Highway Res. Rec. 195, 66-74.