If the coefficient of kinetic friction between tires and dry pavement is 0.800, what is the shortest distance in which an automobile can be stopped by locking the brakes when traveling at 25.7 ?
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To calculate the shortest stopping distance for an automobile, we need to consider the laws of physics and use the information provided. Here's how you can find the answer:
1. Gather the given information:
- Coefficient of kinetic friction (μ) = 0.800
- Initial velocity (v) = 25.7 m/s
2. Determine the acceleration:
The formula for acceleration (a) is given by Newton's second law of motion: F = m × a, where F is the force of friction and m is the mass of the automobile. However, we want to find the acceleration caused by friction, so we rearrange the formula as: a = F / m.
The force of friction (F) can be calculated using the formula: F = μ × N, where N is the normal force exerted on the tires by the pavement. In this case, the normal force is equal to the weight of the automobile, which is given by the formula: N = m × g, where g is the acceleration due to gravity (9.8 m/s²).
Combining these formulas, we get: a = (μ × m × g) / m.
Simplifying further, we have: a = μ × g.
Substituting the values: a = 0.800 × 9.8 = 7.84 m/s².
3. Calculate the stopping distance:
To find the stopping distance, we can use the equation of motion: v² = u² + 2as, where u is the initial velocity, v is the final velocity (which is 0 in this case), a is the acceleration, and s is the stopping distance.
Rearranging the formula, we get: s = (v² - u²) / (2a).
Substituting the values: s = (0 - (25.7)²) / (2 × (-7.84)).
Calculating further, we find: s = (0 - 660.49) / (-15.68) = 42.081 m.
So, the shortest distance in which the automobile can be stopped by locking the brakes when traveling at 25.7 m/s is approximately 42.081 meters.