YELLOW LIGHT PROBLEM SOLUTION:

BRAKING distance to be farther away to safely brake

ACCELERATING the distance to be safely away or closer is

1)for never reach speed limit) (ACADEMIC)

-----------------------------------------------------------------------------------(HONORS ONLY)-------------

2)for (reacting, accelerating up to speed limit, at speed limit)

-----------------------------------------------------------------------------------(HONORS ONLY)-------------

3)for ( already over speed limit)

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Distance

Away Da

Velocity

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YELLOW LIGHT BRAKING CASE:

A_B is your braking acceleration (a negative number) of car in m/s² gotten by using . Braking is usually given as 60mph-0 or 40mph-0 in distance to stop in feet.

So

If there is no reaction time then the distance to brake to a stop uses ,

so distance to be farther away to safely brake

is so

If you figure in reaction time, then the distance traveled is the distance you go in that reaction time (at a constant speed) plus the distance you go while braking to a stop.

so so

distance to be farther away to safely brake

YELLOW LIGHT ACCELERATING CASE

A = acceleration of car in m/s² gotten by where is known in sec.

YELLOW LIGHT ACCELERATING CASE

If no reaction time or no speed limit, then the distance traveled in the time of the yellow light is simply:

so then the distance to be safely away or closer is so or

YELLOW LIGHT ACCELERATING CASE

If you figure in reaction time, then the distance traveled in the time of the yellow light is the distance you go in that reaction time (at a constant speed) plus the distance you go while accelerating in the time left.

so

so then the distance to be safely away or closer is

(ACADEMIC---à)

This is also true if you figure in the speed limit if your car is traveling too slow or doesn’t have enough acceleration to accelerate up to the speed limit in the time leftover from the yellow light.

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If your car is going faster than that, then it goes up to the speed limit and then have to stay at that speed.

That critical velocity of is (using with and ) is

When the car’s initial speed is at or above the speed limit, ()it simply stays the same speed for the entire time of the yellow light so that the distance is

and

so then the distance to be safely away or closer is

Technically, the car is already illegal anyway (over the speed limit), so it wouldn’t make sense to have it drop down to the speed limit as it goes through the yellow light.

YELLOW LIGHT ACCELERATING CASE

If the car’s initial speed is over the critical velocity (),but less than the speed limit ()it will be three parts, the distance you go in your reaction time at a constant speed, plus the distance while accelerating up to the speed limit, plus the distance you go while at the speed limit in the time you have left.

-The distance in reaction time:

-The distance while accelerating from to using

or solving for the time it takes to accelerate to the speed limit. Using ,

using , and substituting for ,

which simplifies to:

-The distance the car goes at the speed limit is the velocity at the speed limit times the time left.

The time left is the time of the yellow light minus the reaction time minus the time spent accelerating up to the speed limit. =

So =

To get the total distance in this case: when the car’s initial speed is over that amount (),but less than the speed limit ()

,

so then the distance to be safely away or closer is

simplified a bit:

for

YELLOW LIGHT PROGRAM

FOR TI-82/83/84

ClrHome

ClrTable

Clear Entries

ClrTable

ClrDraw

Disp ""

Disp "YELLOW LIGHT"

Disp "PROBLEM!!!"

Pause

ClrHome

Disp "CAR LENGTH IN M?"

Disp ""

Disp "(TYPICAL VALUE"

Disp " IS 3 M) "

Input L

ClrHome

Disp ""

Disp "CAR BRAKING"

Disp "DISTANCE"

Disp "60MPH TO 0"

Disp " IN FEET?"

Disp ""

Disp "(TYPICAL VALUE"

Disp " IS 140 FT) "

Input B

"COMMENTBraKInG DeacceLeratIOn In M/s 0 Is"

(60*.44704)0/(2*(B*.305))→C

ClrHome

Disp "CAR ACCELERATION"

Disp " O TO 60 MPH"

Disp " IN SECONDS?"

Disp ""

Disp "(TYPICAL VALUE"

Disp " IS 7-15 SEC "

Input A

"COMMENTAcceLeratIOn OF Car In M/s 0 Is"

(60*.44704)/A→F

ClrHome

Disp "REACTION TIME "

Disp "IN SECONDS?"

Disp ""

Disp "(TYPICAL VALUE"

Disp "IS .2 TO 1 S) "

Input R

ClrHome

Disp "TIME OF YELLOW"

Disp "LIGHT IN SECONDS?"

Disp ""

Disp "(TYPICAL VALUE"

Disp " IS 2-8) "

Input T

ClrHome

Disp "INTERSECTION "

Disp "LENGTH IN METERS?"

Disp ""

Disp "(TYPICAL VALUE"

Disp " IS 8-20 M) "

Input I

ClrHome

Disp "SPEED LIMIT(MPH)"

Disp ""

Disp "(TYPICAL VALUE"

Disp " 25-50 MPH) "

Input S

"COMMENTMAX InItaL speed In MpH tHe car can GO MaX tO speed up tO tHe speed LIMIt In tHIs tIMe Is"

(S-60*(T-R)/A)→Z

"COMMENTTime In sec It taKes tHe car tO GO FrOM Its InItIaL speed tO tHe speed LIMIt Is"

"(A*(S-X)/60)"→Y₆

ClrHome

Disp "VELOCITY(MPH)"

Disp "OF CAR"

Disp "ON X AXIS"

Disp ""

Disp "DISTANCE(M)"

Disp "FROM LIGHT"

Disp "ON Y AXIS"

Pause

–9→Xmin

85→Xmax

5→Xscl

1→Xres

–50→Ymin

Y₁(85)→Ymax

5→Yscl

Disp "Y1 IS BRAKING"

Disp " CASE >"

Disp "Y2 IS SPEED UP"

Disp " CASE <"

Pause

"COMMENTX Is InItIaL veLOcItY varIabLe OF car In MpH"

"COMMENTDIstance In M TraveLed In reactIOn tIMe Is"

".44704*X*R"→Y₇

"COMMENTBraKInG Case, DIstance In M traveLed In braKInG deacceLeratIOn Is"

"(.44704*X)0/(2*C)"→Y₈

"COMMENTBraKInG Case, DIstance traveLed In reactIOn tIMe pLus deacceLeratIOn Is"

"Y₇+Y₈"→Y₁

"COMMENTAcceLeratInG case IF nO speed LIMIt, reactIOn tIMe dIstance In M pLus dIstance In M traveLed as acceLeratInG MInus IntersectIOn MInus car LenGtH Is"

"Y₇+(.44704*X)*(T-R)+.5*F*(T-R)0-I-L"→Y₃

"COMMENTReactIOn tIMe dIstance pLus dIstance traveLed In M wHILe acceLeratInG up tO speed LIMIt pLus tHe dIstance traveLed aFter tHat wHILe at tHe speed LIMIt MInus IntersectIOn MInus car LenGtH Is"

"Y₇+(.44704*X)*Y₆+.5*F*Y₆0+.44704*S*(T-R-Y₆)-I-L"→Y₄

"COMMENTDIstance traveLed IF at or abOve speed LIMIt MInus IntersectIOn MInus car LenGtH (reactIOn tIMe dOesn't Matter, assuMe a cOnstant speed above tHe speed LIMIt) Is"

"(.44704*X)*T-I-L"→Y₅

"COMMENTTHree part FunctIOn FOr acceLeratIOn case. LIne FOr InItIaL speed Less tHan MaX InItIaL speed, pLus LIne FOr tHe InItIaL speed Is In between MaX InItIaL speed and speed LIMIt, and LIne FOr car's InItIaL speed Is at or abOve speed LIMIt Is"

"(Y₃)*(X<Z)+(Y₄)*(X≥Z)*(X<S)+Y₅*(X≥S)"→Y₂

PlotsOff

LabelOn

RectGC

AxesOn

CoordOn

GridOff

FnOff

FnOn 1,2,3

GraphStyle(1,3)

"COMMENTGrapH's BRaKInG Case as a Greater tHan"

GraphStyle(2,4)

"COMMENTGrapHs AcceLeratIOn Case as a Less tHan"

GraphStyle(3,5)

"COMMENTGrapHs nO speed LIMIt case"

Vertical X

Trace

ClrHome

Fix 1

Disp "ENTER SPEED OF"

Disp "CAR IN MPH"

Input V

Disp "BRAKE IF"

Disp "FARTHER THAN"

Disp Y₁(V)

Disp "METERS. SPEED UP"

Disp "IF CLOSER THAN"

Disp Y₂(V)," METERS AWAY"

Pause

ClrHome

Disp "THINKING"

0→X

If Y₂(0)<Y₁(0)

Then

Disp "NO SOLUTION!"

Else

solve((Y₁-Y₂),X,55,{-1000,1000})→X

End

Disp X," MPH,"

Disp

Disp Y₁(X)," METERS AWAY"

Disp "IS THE DECISION"

DIsp" POINT"

Pause