Traveling at a speed of 20.0 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 0.540. What is the speed of the automobile after 1.11 s have elapsed? Ignore the effects of air resistance.
To calculate the speed of the automobile after 1.11 s have elapsed, we can use the equation of motion:
v = u + at
where:
v = final velocity
u = initial velocity
a = acceleration
t = time
In this case, the initial velocity (u) is 20.0 m/s, and the time elapsed (t) is 1.11 s.
Now, let's calculate the acceleration (a) using the equation:
a = -μg
where:
μ = coefficient of kinetic friction
g = acceleration due to gravity (9.8 m/s^2)
In this case, the coefficient of kinetic friction (μ) is 0.540.
Now, substituting the values into the equation, we get:
a = -0.540 * 9.8
a = -5.292 m/s^2
Now, let's calculate the final velocity (v) using the equation:
v = u + at
Substituting the values into the equation, we get:
v = 20.0 + (-5.292 * 1.11)
v = 20.0 - 5.878
v = 14.122 m/s
Therefore, the speed of the automobile after 1.11 s have elapsed is 14.122 m/s.
To find the speed of the automobile after 1.11 seconds have elapsed, we need to use the equation for acceleration:
\[ a = \frac{{v_f - v_i}}{{t}} \]
Where:
- \( a \) is the acceleration
- \( v_f \) is the final velocity
- \( v_i \) is the initial velocity
- \( t \) is the time taken
In this case, the initial velocity (\( v_i \)) is 20.0 m/s, the time taken (\( t \)) is 1.11 seconds, and we need to find the final velocity (\( v_f \)).
To calculate the acceleration, we need to first determine the net force acting on the automobile. The net force can be calculated using the equation:
\[ f_{net} = \mu_k \cdot f_N \]
Where:
- \( f_{net} \) is the net force
- \( \mu_k \) is the coefficient of kinetic friction
- \( f_N \) is the normal force
The normal force (\( f_N \)) is the force exerted by the road on the tires and is equal to the weight of the automobile, which can be calculated as:
\[ f_N = m \cdot g \]
Where:
- \( m \) is the mass of the automobile
- \( g \) is the acceleration due to gravity (approximately 9.8 m/s²)
Once we have the net force, we can calculate the acceleration using Newton's second law:
\[ f_{net} = m \cdot a \]
Now let's plug in the values and solve the problem step by step:
Step 1: Calculate the normal force
The mass of the automobile is not provided in the question, so let's assume it is 1000 kg.
\[ f_N = m \cdot g = 1000 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 9800 \, \text{N} \]
Step 2: Calculate the net force
\[ f_{net} = \mu_k \cdot f_N = 0.540 \times 9800 \, \text{N} = 5292 \, \text{N} \]
Step 3: Calculate the acceleration
\[ f_{net} = m \cdot a \]
\[ a = \frac{{f_{net}}}{{m}} = \frac{{5292 \, \text{N}}}{{1000 \, \text{kg}}} = 5.292 \, \text{m/s}^2 \]
Step 4: Calculate the final velocity
\[ a = \frac{{v_f - v_i}}{{t}} \]
\[ v_f = (a \cdot t) + v_i = (5.292 \, \text{m/s}^2 \times 1.11 \, \text{s}) + 20.0 \, \text{m/s} = 25.863 \, \text{m/s} \]
Therefore, the speed of the automobile after 1.11 seconds have elapsed is approximately 25.863 m/s.