A car is traveling at 104 km/h when the driver sees an accident 50 m ahead and slams on the brakes. What minimum constant deceleration is required to stop the car in time to avoid a pileup

a =  - 8.34 m/sec²       ( deceleration or negative)  

Equations for UAM  ( uniformly accelerated motion) are:

vf = v₀ ±  a*t           and      s  =  s₀  + v₀*t + (1/2)*a*t²

In our case, the motion is with deceleration, then

vf = v₀ - a*t       and    s = s₀  +  v₀*t  - (1/2)*a*t²  

working on these equatios we get:

vf = v₀ - a*t     (1)           s  -  s₀   =  v₀*t  - (1/2)*a*t²      (2)

v₀ - vf = a*t

t  =  (v₀ - vf)/a

By substitution of (1) in equation (2)

s  -  s₀   =  v₀ * (v₀ - vf)/a  -  (1/2) * a* [(v₀ - vf)/a]²

s  -  s₀   =  (v₀² - v₀*vf)/a  -   (1/2) * a* (1/a²)* (v₀ - vf)²

s  -  s₀   =  1/a * ( v₀² - v₀*vf ) - 1/a* (1/2)  * (v₀ - vf)²

s  -  s₀   =  1/a* [  ( v₀² - v₀*vf ) - (1/2) * (v₀ - vf)²]

a * (s  -  s₀ )  =   v₀² - v₀*vf - v₀²/2 - vf²/2 + v₀*vf

a * (s  -  s₀ )  =  (1/2) * v₀² - (1/2)*vf²

a * (s  -  s₀ )  =  (1/2) * ( v₀² - vf²)

We find an expression to calculate the minimum deceleration to stop the car in time to avoid crashing

s₀ = 50 meters            s  =  0        v₀ =  104 Km/h    vf = 0

1 Km  = 1000 m    and   1 h = 3600 sec

v₀ = 104 Km/h    =  28.88 m/sec

a  =  (1/2) [ (28.88)² - 0 ] / 0 - 50

a =  - 8.34 m/sec²       ( deceleration or negative)  

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