A car traveling at 20 m/s applies the brakes. The coefficients of static and kinetic friction between the tires

and the road are 0.80 and 0.50 respectively.
(A) Determine the stopping distance if the car is able to
stop as quickly as possible without skidding.
(B) Determine the stopping distance if the car skids.

A. 25.5m

B. 40.8m

(A) To determine the stopping distance without skidding, we need to find the maximum static friction between the tires and the road. The maximum static friction can be calculated using the formula:

F_static_max = μ_static * N

where F_static_max is the maximum static friction, μ_static is the coefficient of static friction, and N is the normal force.

The normal force is equal to the weight of the car, which 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 (approximately 9.8 m/s^2).

Given that the coefficient of static friction is 0.80, we can calculate the maximum static friction:

F_static_max = 0.80 * m * g

Next, we need to find the deceleration of the car when the brakes are applied. The deceleration can be calculated using the formula:

a = F_net / m

where a is the deceleration, F_net is the net force acting on the car, and m is the mass of the car.

When the car is stopping, the net force acting on the car is equal to the frictional force. Therefore,

F_net = F_friction = F_static_max

We can now calculate the deceleration:

a = F_static_max / m

Finally, we can use the kinematic equation to find the stopping distance:

v^2 = u^2 + 2as

where v is the final velocity (0 m/s), u is the initial velocity (20 m/s), a is the deceleration, and s is the stopping distance.

Plugging in the values:

0 = (20 m/s)^2 + 2 * a * s

Simplifying this equation, we find:

s = (20 m/s)^2 / (2 * a)

Substituting the value for a, we can solve for s:

s = (20 m/s)^2 / (2 * (0.80 * m * g))

(B) To calculate the stopping distance if the car skids, we need to use the coefficient of kinetic friction.

The kinetic frictional force can be calculated using the formula:

F_kinetic = μ_kinetic * N

where F_kinetic is the kinetic frictional force and μ_kinetic is the coefficient of kinetic friction. The normal force N is the same as before.

The net force acting on the car is now equal to the kinetic frictional force:

F_net = F_kinetic

Using the equation for deceleration, we find:

a = F_kinetic / m

We can then use the same kinematic equation as before to find the stopping distance:

0 = (20 m/s)^2 + 2 * a * s

Simplifying this equation, we find:

s = (20 m/s)^2 / (2 * a)

Substituting the value for a using the new coefficient of kinetic friction, we can solve for s.

To determine the stopping distance, we need to consider two scenarios: when the car stops without skidding and when it skids.

(A) Stopping distance without skidding:
When the car stops without skidding, the maximum friction force that can be applied is the static friction force. The coefficient of static friction is 0.80.

To calculate the maximum static friction force, we use the formula:
Friction force = (Coefficient of static friction) * (Normal force)

The normal force is equal to the weight of the car, which can be calculated as:
Normal force = (Mass of the car) * (Acceleration due to gravity)

Once we have the friction force, we can calculate the deceleration of the car using Newton's second law:
Force = Mass * Acceleration

Since the car is traveling at 20 m/s initially and comes to a stop, the final velocity is 0 m/s.

We can find the time it takes to stop by using the equation of motion:
Final velocity = Initial velocity + (Acceleration * Time)

Once we have the time, we can calculate the stopping distance using the equation of motion:
Stopping distance = (Initial velocity * Time) + (0.5 * Acceleration * Time^2)

(B) Stopping distance with skidding:
When the car skids, the friction force is the kinetic friction force. The coefficient of kinetic friction is 0.50.

Using the same procedure as before, we calculate the deceleration, time, and stopping distance.

By following these steps, we can find the stopping distance for both scenarios: with and without skidding.

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