An automobile's wheels are locked as it slides to a stop from 34.9 m/s. If the coefficient of kinetic friction is 0.266 and the road is horizontal, how long does it take the car to stop?
To find the time it takes for the car to stop, we need to use the equation of motion that relates the final velocity, initial velocity, time, and acceleration. In this case, the acceleration is due to the frictional force acting on the car.
We can start by calculating the acceleration of the car. The frictional force can be determined using the formula:
frictional force = coefficient of kinetic friction * normal force
The normal force acting on the car is equal to the weight of the car, which can be calculated using the formula:
normal force = mass * gravity
where mass is the mass of the car and gravity is the acceleration due to gravity.
Once we have the frictional force, we can calculate the acceleration using Newton's second law:
frictional force = mass * acceleration
Rearranging this equation, we get:
acceleration = frictional force / mass
With the acceleration known, we can now use the equation of motion:
final velocity = initial velocity + acceleration * time
Given that the initial velocity is 34.9 m/s and the final velocity is 0 m/s (since the car comes to a stop), we can rearrange the equation to solve for time:
time = (final velocity - initial velocity) / acceleration
Now, let's calculate the time it takes for the car to stop:
1. Determine the normal force:
- Find the weight of the car using its mass (given or known value) and the acceleration due to gravity (approximately 9.8 m/s^2).
- Calculate the normal force using the formula normal force = mass * gravity.
2. Calculate the frictional force:
- Use the given coefficient of kinetic friction and the normal force calculated in step 1.
- Calculate the frictional force using the formula frictional force = coefficient of kinetic friction * normal force.
3. Determine the acceleration:
- Divide the frictional force obtained in step 2 by the mass of the car to find the acceleration.
4. Calculate the time:
- Subtract the initial velocity (34.9 m/s) from the final velocity (0 m/s).
- Divide the result by the acceleration obtained in step 3 to find the time it takes for the car to stop.