Given, speed of the car as well as truck = 72 km/h

$=72\times \frac{5}{18}\mathrm{m}/\mathrm{s}=20\mathrm{m}/\mathrm{s}$

$\text{ Retarded motion for truck }\mathrm{v} = \mathrm{u} + {\mathrm{a}}_{\mathrm{t}}\mathrm{t}$
$0 = 20 + {\mathrm{a}}_{\mathrm{t}} \times 5$
$\mathrm{Or} {\mathrm{a}}_{\mathrm{t}} = - 4\mathrm{m}/{\mathrm{s}}^2$
$\mathrm{Retarded} \mathrm{motion} \mathrm{for} \mathrm{the} \mathrm{car} \mathrm{v}=\mathrm{u}+{\mathrm{a}}_{\mathrm{c}}\mathrm{t}$
$0 = 20 + {\mathrm{a}}_{\mathrm{c}} \times 3$
$\mathrm{Or} {\mathrm{a}}_{\mathrm{c}} = -\frac{20}{3}\mathrm{m}/{\mathrm{s}}^2$

Step 2: Find velocities of car and truck after time t.

Let the car be at a distance x from the truck when the truck gives the signal and t be the time taken to cover this distance.

As human response time is 0.5 s, therefore, the time of retarded motion of the car is (t - 0.5) s.

The velocity of the car after time t,

$\begin{array}{l}{v}_c=u-at\\ =20-\left(\frac{20}{3}\right)(t-0.5)\end{array}$

The velocity of the truck after time t,

$v_t=20-4t$

Step 3: To avoid the car bump onto the truck,

$\begin{array}{c}{\mathrm{v}}_{\mathrm{c}}={\mathrm{v}}_{\mathrm{t}}\\ 20-\frac{20}{3}(\mathrm{t}-0.5)=20-4\mathrm{t}\\ 4\mathrm{t}=\frac{20}{3}(\mathrm{t}-0.5)\\ \mathrm{t}=\frac{5}{3}(\mathrm{t}-0.5)\\ 3\mathrm{t}=5\mathrm{t}-2.5\\ \Rightarrow \mathrm{t}=\frac{2.5}{2}=\frac{5}{4}\mathrm{s}\end{array}$

Step 4: Distance traveled by truck in time t,

$\begin{array}{l}{\mathrm{s}}_{\mathrm{t}} = {\mathrm{u}}_{\mathrm{t}}\mathrm{t} + \frac{1}{2}{\mathrm{a}}_{\mathrm{t}}{\mathrm{t}}^2\\ = 20 \times \frac{5}{4} + \frac{1}{2} \times (-4) \times (\frac{5}{4}{)}^2\\ = 21.875\mathrm{m}\end{array}$

Step 5: Distance traveled by car in time t = Distance traveled by car in 0.5 s (without retardation) +Distance traveled by car in (t - 0.5)s (with retardation)

$\begin{array}{l}{\mathrm{s}}_{\mathrm{c}} = (20\times 0.5)+20(\frac{5}{4}-0.5)-\frac{1}{2}\left(\frac{20}{3}\right)(\frac{5}{4}-0.5{)}^2\\ = 23.125\mathrm{m}\\ \mathbf{Step}\mathbf{ }\mathbf{6}\mathbf{:} \mathrm{Find} \mathrm{distance} \mathrm{between} \mathrm{car} \mathrm{and} \mathrm{truck} \mathrm{for} \mathrm{no} \mathrm{collision}.\\ {\mathrm{s}}_{\mathrm{c}}-{\mathrm{s}}_{\mathrm{t}} = 23.125-21.875 = 1.250\mathrm{m}.\end{array}$