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You turn a corner and are driving up a steep hill. Suddenly, a small boy runs out onto the street chasing a ball. You slam on the brakes and skid to a stop, leaving a 50-foot-long skid mark on the street. The boy calmly walks away, but a policeman watching from the sidewalk walks over and gives you a speeding ticket. He points out that the speed limit on this street is 25 mph. After recovering your wits, you begin to examine the situation. You determine that the street makes an angle of 25° with the horizontal and that the coefficient of static friction between your tires and the street is 0.80. You also find that the coefficient of kinetic friction between your tires and the street is 0.60. Your car’s information book tells you that the mass of your car is 1600 kg, and you weigh 140 lbs.

Calculate whether you were speeding based on the given information.

Answer :

Given data:

Total displacement of the car;

[tex]s=50\text{ ft}[/tex]

Speed limit;

[tex]v_m=25\text{ mph}[/tex]

The angle of street from horizontal;

[tex]\theta=25\degree[/tex]

Coefficient of static friction;

[tex]\mu_s=0.80[/tex]

Coefficient of kinetic friction;

[tex]\mu_k=0.60[/tex]

Mass of the car;

[tex]M=1600\text{ kg}[/tex]

Weight of the man;

[tex]W=140\text{ lbs}[/tex]

The kinetic friction force is given as,

[tex]F_k=\mu_k(M+m)g\cos \theta[/tex]

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,

[tex]\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}[/tex]

Substituting all known values,

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

The velocity of the car is given as,

[tex]v^2=u^2-2as[/tex]

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,

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

Substituting all known values,

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

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

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Rewritten by : Barada