A 1000 kg car is traveling at 28 m/s. The driver wishes to stop in a distance of 105 m. What friction between the tires and road is necessary to do...

Hello,
let's solve this problem!

So we have a car travelling at and
we wish to make it stop after braking for a total distance of . We want to know what the
magnitude of the force of friction must be such that this scenario is possible.

Keep in mind we're assuming that the friction between the tires and the
road is constant along the distance.

Our best tool to solve
this problem is the concept of mechanical energy. Our car has a total kinetic energy
of:

where is the mass of the car,
, and is its speed (in meters per second), that is, . Plugging in the
values we get a total kinetic energy of

392000 J

The second concept that we use is that of the work exerted by
a constant force . To make our car stop, we have to apply a certain constant force along
a distance such that the total work done by the force is equal to the total kinetic
energy of the car.

Think like this, our car has mechanical energy, and
this energy must be transformed into heat by the brakes. Transforming mechanical energy into
heat requires work, and this work is done by the force of friction. So the total work done by
the force must be equal to the total energy of our car, since we want to stop the car.

Now, the work done by a constant force applied along a distance is equal
to

But we have that , by the
last calculation, and as given. Plugging in the values and solving for gives the
value of the force:

Thus, the friction between the tires and the road must be approximately
. This is close to the force required to lift a weight!