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 0.650. What is the speed of the automobile after 1.20 s have elapsed? Ignore the effects of air resistance.
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Well, I must say, that driver sure knows how to make a dramatic entrance! Now, let's calculate the speed of the automobile after 1.20 seconds.
First, we need to determine the deceleration of the car. Using the equation of motion, we have:
Final Velocity = Initial Velocity + (Acceleration * Time)
Since the car's initial velocity is 16.1 m/s and the final velocity is what we're trying to find, the equation becomes:
Final Velocity = 16.1 m/s + (Acceleration * 1.20 s)
Now, we need to find the acceleration. The force of friction can be calculated using the formula:
Force of Friction = Coefficient of Kinetic Friction * Normal Force
The normal force is equal to the weight of the car, which can be calculated as:
Normal Force = Mass * acceleration due to gravity
Finally, using Newton's second law of motion, we can find the acceleration:
Force of Friction = Mass * Acceleration
Rearranging the equation, we get:
Acceleration = Force of Friction / Mass
Now, substituting the values we have, we get:
Acceleration = (0.650 * Normal Force) / Mass
Acceleration = (0.650 * (Mass * acceleration due to gravity)) / Mass
Acceleration = 0.650 * acceleration due to gravity
Using the value of acceleration due to gravity as 9.8 m/s^2, we can determine the acceleration as approximately 6.37 m/s^2.
Now, let's go back to our initial equation and substitute the values:
Final Velocity = 16.1 m/s + (6.37 m/s^2 * 1.20 s)
Final Velocity = 16.1 m/s + 7.64 m/s
Final Velocity = 23.74 m/s
So, after 1.20 seconds, with tires screeching and smoke billowing, the speed of the automobile is approximately 23.74 m/s.
To find the speed of the automobile after 1.20 seconds have elapsed, we can use the equation of motion:
v = u + at
Where:
v = final velocity (speed of the automobile after 1.20 s)
u = initial velocity (speed of the automobile before braking)
a = acceleration
t = time
Initially, the speed of the automobile (u) is given as 16.1 m/s. The acceleration (a) can be determined using the force of kinetic friction:
f = μN
Where:
f = force of kinetic friction
μ = coefficient of kinetic friction
N = normal force (equal to the weight of the automobile)
Since there is no vertical motion, the normal force is equal to the weight of the automobile, which can be calculated using the formula:
N = mg
Where:
m = mass of the automobile
g = acceleration due to gravity (approximately 9.8 m/s²)
Now, let's calculate the necessary values step by step:
1. Calculate the normal force (N):
N = mg
= mass × acceleration due to gravity
= mass × 9.8 m/s²
2. Calculate the force of kinetic friction (f):
f = μN
= μ × N
3. Calculate the acceleration (a):
a = -f/m
= -μ × N / m
(Note the negative sign indicates that the acceleration is in the opposite direction to the motion of the automobile)
Now, once we have the acceleration, we can proceed to find the final velocity after 1.20 seconds have elapsed:
v = u + at
Simply plug in the given values and solve for v.
Vx=Ux+at
Vx=16.1-(.65*9.81)(1.2)
Vx=8.4482