The length of the transition curve on the basis of rate of change of acceleration is given as :

\(L = \frac{{{V^3}}}{{RC}}\)

Where

V is the Design Speed (m/sec)

R is the radius of transition curve at point where it meets to circular curve (m)

C is the jerk or rate of change of acceleration. (m/sec^{3})

Further,

The ratio of centrifugal force (P) and weight of vehicle (W) , which is called impact factor, is given as:

\(\frac{P}{W} = \;\frac{{{V^2}}}{{gR}}\)

__Calculation:__

Given:

P/W = 1/4

g = 10 m/s^{2}, V = 20 m/s = 72 kmph

Acceleration a = 0.25 m/s^{2}

Radte of change of centrifugal Acceleration (C) = 80/(75 + V)

where V is in kmph

i.e. (C) = 80/(75 + V) = 80/(75 + 72) = 0.544 m/sec^{3}

We know that \(\frac{P}{w} = \frac{{{V^2}}}{{{g_R}}}\)

\(\frac{1}{4} = \frac{{{{\left( {20} \right)}^2}}}{{10\; \times\; R}} \Rightarrow R = 160\;m\)

Also,

\(L = \frac{{{V^3}}}{{RC}} = \frac{{{{\left( {20} \right)}^3}}}{{160\; \times \;0.54}} = 91.91\;m\)

∴ R = 160 m & L = 92 m is most appropriate answer.