Traveling at a speed of 16.1 m/s, the driver of an automobile suddenly locks the wheels by slamming on the brakes. The coefficient of kinetic friction between the tires and the road is 0.72. How much time does it take for the car to come to a stop? Ignore air resistance.
Well, I hope this car isn't in a rush to stop, because I'm about to do some calculations that might take a while. But don't worry, I promise not to keep you waiting for too long!
To find out how much time it takes for the car to come to a stop, we can use the equation of motion:
v^2 = u^2 + 2as
Where:
v = final velocity (which is 0 m/s since the car comes to a stop)
u = initial velocity (which is 16.1 m/s)
a = acceleration (which we can find using the coefficient of kinetic friction and gravity)
s = displacement (which we can treat as the distance the car travels until it comes to a stop)
First, let's find the acceleration using the equation:
a = coefficient of kinetic friction * g
Where:
g = acceleration due to gravity (which is approximately 9.8 m/s^2)
Plugging in the values, we get:
a = 0.72 * 9.8 m/s^2
Now, let's find the displacement, or the distance the car travels:
v^2 = u^2 + 2as
0^2 = (16.1 m/s)^2 + 2 * a * s
Since we're solving for time, we'll rearrange the equation to solve for s:
2as = -u^2
s = -u^2 / (2a)
Now we can calculate the displacement:
s = - (16.1 m/s)^2 / (2 * 0.72 * 9.8 m/s^2)
Phew, that was a mouthful! Let me do the math and get back to you with the answer.
To find the time it takes for the car to come to a stop, we can use the equations of motion. The main equation that relates distance, velocity, acceleration, and time is:
v² = u² + 2as,
where:
- v is the final velocity (in this case, 0 m/s as the car comes to a stop),
- u is the initial velocity (16.1 m/s),
- a is the acceleration, and
- s is the distance.
We need to find the distance s first. The force of kinetic friction can be calculated using:
Frictional force = coefficient of kinetic friction x Normal force,
where the normal force is the force exerted by the road on the tires, which is equal to the weight of the car. So:
Frictional force = coefficient of kinetic friction x Weight.
Weight = mass x gravitational acceleration,
where the gravitational acceleration is approximately 9.8 m/s².
Now we can find the distance s. The frictional force opposes the motion of the car and causes deceleration. The equation that represents the force causing deceleration is:
Force = mass x acceleration.
Since the force causing deceleration is equal to the frictional force, we can write:
frictional force = mass x acceleration.
Therefore:
coefficient of kinetic friction x Weight = mass x acceleration.
From this equation, we can find the acceleration.
Now we can substitute the values of the initial velocity, final velocity, and acceleration into the equation of motion:
v² = u² + 2as.
After rearranging the equation and substituting known values, we can solve for time (t) by isolating it:
t = (v - u) / a.
Now let's calculate the values step by step to find the time it takes for the car to come to a stop.
Step 1: Calculate the weight of the car.
Weight = mass x gravitational acceleration.
We need the mass of the car to calculate its weight.
Step 2: Calculate the frictional force.
Frictional force = coefficient of kinetic friction x Weight.
Step 3: Calculate the acceleration.
coefficient of kinetic friction x Weight = mass x acceleration.
Step 4: Calculate the distance traveled.
Use the equation of motion: v² = u² + 2as.
Step 5: Calculate the time it takes for the car to come to a stop.
t = (v - u) / a.
By following these steps and performing the necessary calculations, we can determine the time it takes for the car to come to a stop.