Traffic police examine the scene of an accedient involving two cars and measure a 72 metre long skid mark of one of the cars, which nearly came to stop before colliding.The accdient occured on a section of highway that had a 100 kilometre per hour speed limit. The coefficient of kinetic friction between rubber from tyres and the pavement is about 0.80. The driver of the vehicle that created the long skid marks claims to have been travelling at less than 100 kilometre per hour. Is he telling the truth?
Fc = M*g = 9.8M = force of the car = Normal force(Fn).
Ff = u*Fn = 0.8*9.8M = 7.84M = force of kinetic friction.
Fp = Mg*sinA = mg*sin 0 = 0 = Force parallel with the plane.
Fp-Ff = M*a.
0 - 7.84M = M*a,
divide both sides by M:
a = -7.84 m/s^2.
Vo = 100km/h = 100,000m/3600s = 27.78 m/s.
V^2 = Vo^2 + 2a*d = 0.
27.78^2 + (-15.7)d = 0,
d = 49 m = stopping distance when driving 100km/h.
The driver who created the 72-meter skid marks was driving above the speed limit.
To determine if the driver is telling the truth about their speed, we can use the concept of skid marks and the coefficient of kinetic friction.
First, we need to find the initial velocity (speed) of the vehicle. This can be done using the following formula of motion:
v^2 = u^2 + 2as
where v is the final velocity (0 m/s as the car nearly came to a stop), u is the initial velocity (unknown), a is the acceleration (due to friction), and s is the distance (72 meters).
Since the car almost came to a stop, we can assume that the acceleration is approximately equal to the deceleration due to friction. The coefficient of kinetic friction (μ) relates the frictional force (F) to the normal force (N) on the car:
F = μN
The normal force N is equal to the weight of the car (mg), where m is the mass of the car and g is the acceleration due to gravity (9.8 m/s^2). This can be simplified to:
F = μmg
The frictional force can be related to the acceleration using Newton's second law (F = ma), which gives:
μmg = ma
Since the mass (m) cancels out on both sides, we have:
a = μg
Finally, substituting the values into the first formula:
v^2 = u^2 + 2a(-s)
0 = u^2 - 2μg s
Now, we can solve for the initial velocity (u) of the car:
u = sqrt(2μgs)
Substituting the given values:
u = sqrt(2*0.8*9.8*72)
After calculating the value, we find:
u ≈ 28.170 m/s
To convert this velocity to kilometers per hour:
u = 28.170 m/s * (3600 s/1 hr) * (1 km/1000 m)
u ≈ 101.4 km/hr
Based on the calculations, the initial velocity of the car is approximately 101.4 km/hr, which is greater than the reported speed limit of 100 km/hr. Therefore, the driver's claim of traveling at less than 100 km/hr is false.