If the coefficient of kinetic friction between tires and dry pavement is 0.800, what is the shortest distance in which an automobile can be stopped by locking the brakes when traveling at 25.6m/s ?
Wext = KEfinal - KEinitial
f*s = 1/2mVfinal^2 - 1/2 mVinitial ^2
umg*s = 1/2m (Vfinal^2 - Vinitial ^2)
the mass can be cancel and Vfinal is zero since it will stop. So the displacmennt is given by
s = 1/2(Vinitial^2)/u*g
To determine the shortest distance in which the automobile can be stopped by locking the brakes, we can use the concept of kinetic friction.
The force of kinetic friction can be calculated using the formula:
Fk = μk * N
Where:
Fk is the force of kinetic friction
μk is the coefficient of kinetic friction
N is the normal force (equal to the weight of the object)
In this case, the normal force acting on the automobile is equal to its weight, given by:
N = m * g
Where:
m is the mass of the automobile
g is the acceleration due to gravity (approximately 9.8 m/s^2)
The force of kinetic friction can be used to determine the acceleration of the automobile using Newton's second law:
Fk = m * a
From this equation, we can isolate the acceleration:
a = Fk / m
Now, using the kinematic equation:
v^2 = u^2 + 2 * a * s
Where:
v is the final velocity (which is 0 in this case, as the automobile comes to a stop)
u is the initial velocity (25.6 m/s)
a is the acceleration (determined using Fk / m)
s is the distance traveled (which we want to find)
Rearranging the equation to solve for s:
s = (v^2 - u^2) / (2 * a)
Now let's plug in the given values:
μk = 0.800
u = 25.6 m/s
m = (let's assume 1000 kg)
g = 9.8 m/s^2
First, calculate the normal force:
N = m * g
N = 1000 kg * 9.8 m/s^2
N = 9800 N
Next, calculate the force of kinetic friction:
Fk = μk * N
Fk = 0.800 * 9800 N
Fk = 7840 N
Then, calculate the acceleration:
a = Fk / m
a = 7840 N / 1000 kg
a = 7.840 m/s^2
Finally, calculate the shortest stopping distance:
s = (v^2 - u^2) / (2 * a)
s = (0 - (25.6 m/s)^2) / (2 * (7.840 m/s^2))
s = (0 - 655.36 m^2/s^2) / 15.68 m/s^2
s = -655.36 m^2/s^2 / 15.68 m/s^2
s ≈ -41.773 m^2
The negative sign indicates that the distance is in the opposite direction of motion. Since distance cannot be negative in this context, we take the absolute value, giving us:
s ≈ 41.773 meters
Therefore, the shortest distance in which the automobile can be stopped by locking the brakes is approximately 41.773 meters.
To find the shortest stopping distance, we need to consider the forces acting on the automobile. When the brakes are locked, the frictional force between the tires and pavement opposes the motion of the car and brings it to a stop. The frictional force can be calculated using the equation:
frictional force = coefficient of kinetic friction * normal force
The normal force is the force exerted by the surface on the object and is equal to the weight of the automobile (mass * acceleration due to gravity) when it is on a level surface. Since the car is on a level surface, the normal force is equal to the weight.
So, let's calculate the normal force:
mass of the car = m = ?
Given:
coefficient of kinetic friction = 0.800
velocity = 25.6 m/s
We are not given the mass directly, so we cannot calculate the normal force directly. However, we can use the fact that weight is equal to mass times acceleration due to gravity (W = m * g) to find the value of mass.
Weight = m * g
We know the value of acceleration due to gravity (g) is approximately 9.8 m/s².
Weight = m * 9.8
Now, to find the shortest distance in which the car can be stopped, we can use the following equation:
stopping distance = (velocity²) / (2 * acceleration)
The acceleration here is the deceleration due to friction, given by:
acceleration = frictional force / mass
Now, let's calculate the mass of the car:
Weight = m * 9.8
Using the weight, we can find the frictional force:
frictional force = coefficient of kinetic friction * normal force
Once we have the frictional force, we can calculate the acceleration:
acceleration = frictional force / mass
Finally, we can find the stopping distance:
stopping distance = (velocity²) / (2 * acceleration)
Plug in the values and perform the calculations to find the shortest distance in which the car can be stopped.