A 1000 kg car traveling at a speed of 40 m/s skids to a halt on wet concrete where mk = 0.60. How long are the skid marks?
To find the length of the skid marks, we can use the equation of motion that relates the distance traveled during a skid to the initial speed, acceleration, and coefficient of kinetic friction.
First, let's calculate the acceleration of the car. The only force acting on the car during the skid is the force of kinetic friction, which can be determined using the equation:
F_friction = μ_k * m * g
where:
F_friction is the force of friction,
μ_k is the coefficient of kinetic friction (given as 0.60),
m is the mass of the car (given as 1000 kg),
and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the given values, we can calculate the force of friction:
F_friction = 0.60 * 1000 kg * 9.8 m/s^2
= 5880 N
Next, we can use Newton's second law of motion to find the acceleration:
F_net = m * a
where:
F_net is the net force acting on the car,
m is the mass of the car (given as 1000 kg),
and a is the acceleration.
Since the force of friction is the only force acting on the car, we have:
F_friction = m * a
Substituting the known values, we can solve for the acceleration:
5880 N = 1000 kg * a
a = 5880 N / 1000 kg
= 5.88 m/s^2
Now, we can use the equation of motion to find the distance traveled during the skid:
v^2 = u^2 + 2as
where:
v is the final velocity (0 m/s, as the car comes to a halt),
u is the initial velocity (40 m/s),
a is the acceleration (5.88 m/s^2),
and s is the distance traveled during the skid (what we want to find).
Plugging in the given values, we can solve for s:
0^2 = (40 m/s)^2 + 2 * 5.88 m/s^2 * s
0 = 1600 m^2/s^2 + 11.76 m/s^2 * s
-1600 m^2/s^2 = 11.76 m/s^2 * s
s = -1600 m^2/s^2 / 11.76 m/s^2
= -136.05 m
Since distance cannot be negative, the length of the skid marks is 136.05 meters.