A 1900 kg car moves along a horizontal road
at speed v0 = 26.4 m/s. The road is wet, so
the static friction coefficient between the tires
and the road is only μs = 0.17 and the kinetic
friction coefficient is even lower, μk = 0.119.
The acceleration of gravity is 9.8 m/s2 .
What is the shortest possible stopping dis-
tance for the car under such conditions? Use
g = 9.8 m/s2 and neglect the reaction time of
the driver.
a=μ(k)g
s=v²/2a
To find the shortest possible stopping distance for the car, we need to calculate the total distance it takes for the car to stop.
First, let's find the force of static friction (fs) acting on the car:
fs = μs * (mass of the car) * (acceleration due to gravity)
= 0.17 * 1900 kg * 9.8 m/s^2
= 3118.6 N
The force of static friction will act in the opposite direction to the car's motion, which will eventually bring it to a stop.
Next, let's calculate the deceleration of the car using Newton's second law:
Force = mass * acceleration
fs = (mass of the car) * (deceleration of the car)
deceleration = fs / (mass of the car)
= 3118.6 N / 1900 kg
= 1.639 m/s^2
Now, we can calculate the stopping time (t) using the initial velocity (v0) and deceleration (deceleration):
t = v0 / deceleration
= 26.4 m/s / 1.639 m/s^2
≈ 16.1 s
Finally, the shortest possible stopping distance (d) can be calculated using the formula:
d = v0 * t - (1/2) * deceleration * t^2
= 26.4 m/s * 16.1 s - (1/2) * 1.639 m/s^2 * (16.1 s)^2
≈ 212.87 m
Therefore, the shortest possible stopping distance for the car under these conditions is approximately 212.87 meters.
To find the shortest possible stopping distance for the car, we need to consider two different scenarios: when the car is in motion and when it is at rest.
1. When the car is in motion:
In this scenario, the car is experiencing kinetic friction, which opposes its motion. The force of kinetic friction can be calculated using the equation:
Fk = μk * (mass of the car) * (acceleration due to gravity)
where Fk is the force of kinetic friction.
The deceleration of the car due to the force of kinetic friction can be calculated using Newton's second law:
a = Fk / (mass of the car)
The stopping distance can be calculated using the kinematic equation:
stopping distance = (initial velocity^2) / (2 * deceleration)
First, let's calculate the force of kinetic friction:
Fk = (0.119) * (1900 kg) * (9.8 m/s^2)
Fk = 2181.392 N
Now, let's calculate the deceleration:
a = 2181.392 N / (1900 kg)
a ≈ 1.148 m/s^2
Finally, let's calculate the stopping distance:
stopping distance = (26.4 m/s)^2 / (2 * 1.148 m/s^2)
stopping distance ≈ 329.38 m
Therefore, when the car is in motion, the shortest possible stopping distance is approximately 329.38 meters.
2. When the car is at rest:
In this scenario, the car is at rest, so the force of static friction is acting to prevent the car from moving.
The maximum force of static friction can be calculated using the equation:
Fs = μs * (mass of the car) * (acceleration due to gravity)
where Fs is the maximum force of static friction.
Since the car needs to come to rest, the maximum force of static friction must equal the force of kinetic friction:
Fs = Fk
Let's calculate the maximum force of static friction:
Fs = (0.17) * (1900 kg) * (9.8 m/s^2)
Fs = 3197.4 N
Therefore, when the car is at rest, the force of static friction can provide a maximum force of 3197.4 Newtons.
To calculate the stopping distance when the car is at rest, we can use the kinematic equation:
stopping distance = (initial velocity^2) / (2 * deceleration)
Since the initial velocity is zero when the car is at rest, the stopping distance will be zero.
Therefore, when the car is at rest, the stopping distance is zero.
In summary, the shortest possible stopping distance for the car under these conditions is approximately 329.38 meters when the car is in motion and zero when the car is at rest.