Question

You turn a corner and are driving up asteep hill. Suddenly, a small boy runs out on the street chasing a ball. You slam on the brakes and skid to astop leaving a 50-foot-long skid mark on the street. The boy calmly walks away but a policemen watching fromthe sidewalk walks over and gives you a speeding ticket. He points out that the speed limit on this street is 25mph. After you recover your wits, you begin to examine the situation. You determine that the street makes anangle of 25◦with the horizontal and that the coefficient of static friction between your tires and the street is0.80. You also find that the coefficient of kinetic friction between your tires and the street is 0.60. Your car’sinformation book tells you that the mass of your car is 1600 kg. You weigh 140 lbs.

Answer 1

Given data:

Total displacement of the car;

`s=50\text{ ft}`

Speed limit;

`v_m=25\text{ mph}`

The angle of street from horizontal;

`\theta=25\degree`

Coefficient of static friction;

`\mu_s=0.80`

Coefficient of kinetic friction;

`\mu_k=0.60`

Mass of the car;

`M=1600\text{ kg}`

Weight of the man;

`W=140\text{ lbs}`

The kinetic friction force is given as,

`F_k=\mu_k(M+m)g\cos \theta`

Here, m is the mass of the man and g is the acceleration due to gravity.

The acceleration of the car driving up a steep hill is given as,

`\begin{gathered} (M+m)g\sin \theta+F_k=(M+m)a \\ (M+m)g\sin \theta+\mu_k(M+m)g\cos \theta=(M+m)a \\ g\sin \theta+\mu_kg\cos \theta=a \end{gathered}`

Substituting all known values,

`\begin{gathered} (32\text{ ft/s}^2)*\sin (25\degree)+0.6*(32\text{ ft/s}^2)*\cos (25\degree)=a \\ \approx30.92\text{ ft/s}^2 \end{gathered}`

The velocity of the car is given as,

`v^2=u^2-2as`

Here, v is the final velocity (v=0, as the car stops), and u is the initial velocity.

The initial velocity of the car is given as,

`u=\sqrt[]{v^2+2as}`

Substituting all known values,

`\begin{gathered} u=\sqrt[]{0^2+2*(30.92\text{ ft/s}^2)*(50\text{ ft})} \\ \approx55.61\text{ ft/s} \\ \approx37.91\text{ mph} \end{gathered}`

Therefore, your speed is greater than the speed limit. Thus, you can not fight the ticket in the court.