You're driving along at$25m/s$with your aunt's valuable antiques in the back of your pickup truck when suddenly you see a giant hole in the road $55$ ahead of you. Fortunately, your foot is right beside the brake and your reaction time is zero!

a. Can you stop the truck before it falls into the hole?

b. If your answer to part a is yes, can you stop without the antiques sliding and being damaged? Their coefficients of friction are ${\mu }_{\mathrm{s}}=0.60$and${\mu }_{\mathrm{k}}=0.30$.

The initial speed of the truck: $25\mathrm{m}/\mathrm{s}$.

Distance of the hole from moving truck: $55\mathrm{m}$.

Reaction time : $0\mathrm{s}$.

$v^2=u^2+2as$.

In order to determine whether we can stop the truck before it falls into the hole, we have to use the equation of the kinematics $v^2=u^2+2as$

The truck comes to rest, so its final speed be$0\mathrm{m}/\mathrm{s}$.

Now substituting the various values to the equation,

We get,

$\begin{array}{r}{v}^2=u^2+2as\\ 0=(25{)}^2-2(a)55\end{array}$

Here, the$a^2$is the rate of retardation.

Now solving for the value of $a$,

We get

$\begin{array}{r}0=(25{)}^2-2(a)55\\ a=\frac{625}{110}=5.68\mathrm{m}/{\mathrm{s}}^2\end{array}$

So, for the truck to stop before it falls into the hole, the minimum retardation or de-acceleration required is$5.68\mathrm{m}/{\mathrm{s}}^2$.

The initial speed of the truck: $25\mathrm{m}/\mathrm{s}.$

Coefficient of static friction ${\mu }_s:0.60$.

Coefficient of kinetic friction ${\mu }_k:0.30$.

$\begin{array}{rl}& F={\mu }_{s}N\\ & v^2=u^2+2as\end{array}$.

The static friction force can be given as $F={\mu }_{s}N$.

Further, we know that the force is given as $F=ma$.

By equating both the equations as,

$\begin{array}{c}ma={\mu }_{s}N\\ \text{or,}ma={\mu }_{x}mg\\ \text{or,}a={\mu }_{s}g\end{array}$

This value of $a^2$will give the rate of retardation for which the truck stop without the antiques sliding and being damaged.$\begin{array}{rl}& \text{So,}a={\mu }_{s}g\\ & a=0.60\times 9.8=5.88\mathrm{m}/{\mathrm{s}}^2\end{array}$

If we de-accelerate the truck with $5.88/{\mathrm{s}}^2$, the truck can stop without the antiques sliding and being damaged.