A 1600 kg car moves along a horizontal road
at speed v0 = 13.3 m/s. The road is wet, so
the static friction coefficient between the tires
and the road is only μs = 0.21 and the kinetic
friction coefficient is even lower, μk = 0.147.
The acceleration of gravity is 9.8 m/s2 .
Assume: No aerodynamic forces; g =
9.8 m/s2, forward is the positive direction.
What is the highest possible deceleration of
the car under such conditions?
Answer in units of m/s2
5.394090
To find the highest possible deceleration of the car under the given conditions, we need to determine the maximum frictional force that can act on the car.
The maximum static frictional force (F_static_max) can be calculated using the formula:
F_static_max = μs * N
where μs is the static friction coefficient and N is the normal force acting on the car.
The normal force (N) acting on the car can be calculated using the formula:
N = m * g
where m is the mass of the car and g is the acceleration due to gravity.
Substituting the given values, we have:
N = 1600 kg * 9.8 m/s^2
N = 15,680 N
Substituting the values for the static friction coefficient and the normal force into the formula for maximum static frictional force, we have:
F_static_max = 0.21 * 15,680 N
F_static_max = 3,290.8 N
Therefore, the maximum static frictional force that can act on the car is 3,290.8 N.
Since the kinetic friction coefficient (μk) is lower than the static friction coefficient (μs), the kinetic friction force (F_kinetic) will come into play when the car is already in motion.
The maximum kinetic frictional force (F_kinetic_max) can be calculated using the formula:
F_kinetic_max = μk * N
Substituting the given values, we have:
F_kinetic_max = 0.147 * 15,680 N
F_kinetic_max = 2,300.16 N
Therefore, the maximum kinetic frictional force that can act on the car is 2,300.16 N.
Since the deceleration of the car is equal to the net force acting on it divided by its mass (a = F_net / m), the highest possible deceleration of the car can be determined by finding the maximum of the two frictional forces and dividing it by the mass of the car.
The highest possible deceleration (a_max) can be calculated using the formula:
a_max = max(F_static_max, F_kinetic_max) / m
Substituting the given value for the mass of the car, we have:
a_max = max(3,290.8 N, 2,300.16 N) / 1600 kg
a_max = 3,290.8 N / 1600 kg
Performing the calculation, we find:
a_max ≈ 2.055 m/s^2
Therefore, the highest possible deceleration of the car under the given conditions is approximately 2.055 m/s^2.