To find the acceleration of the car due to friction and the distance it slides to a stop, we can follow these steps:
- Calculate the force of friction.
The force of kinetic friction (\( F_f \)) can be calculated using the formula: \[ F_f = \mu_k \cdot F_n \] where:
- \( \mu_k \) is the coefficient of kinetic friction (0.88),
- \( F_n \) is the normal force, which for a car on level ground is equal to its weight (\( \text{mass} \times g \)), where \( g \) is approximately \( 9.81 , \text{m/s}^2 \).
Given:
- Mass of the car (\( m \)) = 1200 kg
- \( g \) = 9.81 m/s²
Calculating the normal force (\( F_n \)): \[ F_n = m \cdot g = 1200 , \text{kg} \cdot 9.81 , \text{m/s}^2 = 11772 , \text{N} \]
Now, calculate the force of kinetic friction: \[ F_f = \mu_k \cdot F_n = 0.88 \cdot 11772 , \text{N} = 10359.36 , \text{N} \]
- Determine the acceleration due to friction.
The acceleration (\( a \)) can be found using Newton's second law: \[ F = m \cdot a \] Here, the friction force will act in the opposite direction to the motion, so we set it as negative: \[ -F_f = m \cdot a \quad \Rightarrow \quad a = -\frac{F_f}{m} \]
Plugging in the values: \[ a = -\frac{10359.36 , \text{N}}{1200 , \text{kg}} = -8.05 , \text{m/s}^2 \]
- Calculate the distance the car slides to a stop.
To find the distance, we can use the kinematic equation: \[ v^2 = u^2 + 2a s \] where:
- \( v \) = final velocity (0 m/s, since the car stops),
- \( u \) = initial velocity (unknown, but it’ll cancel out),
- \( a \) = acceleration (-8.05 m/s²),
- \( s \) = distance.
Rearranging and solving for distance \( s \): \[ 0 = u^2 + 2(-8.05)s \] \[ u^2 = 16.1s \] \[ s = \frac{u^2}{16.1} \]
The distance depends on the initial speed (\( u \)). Assuming you know the initial velocity, you can compute the distance accordingly.
Summary:
- The car's acceleration due to friction is \( a = -8.05 , \text{m/s}^2 \).
- The distance \( s \) to stop can be calculated based on the initial speed \( u \) using the equation \( s = \frac{u^2}{16.1} \).
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