A car with passengers has a mass M=2000 kg and travels with a speed of 65 miles/hour.
Suddenly, the driver the brakes to avoid a collision with another car traveling at 20 miles/hour
at a distance D=30 meters away (98.4ft). Analyze the problem from the moment when the
breaks are fully engaged and tires are not rotating.
Question: will there be collision between the two cars?
Talk about the friction and how would you determine if there will be collision.
You need to solve the problem as a function of the friction coefficient , then discuss your
result based on the following coefficient’s values : {0.1, 0.8}. What should be the minimum safe
distance between the two cars [assuming two values for the coefficient {0.1, 0.8}?
The solution needs to include the following:
1. All diagrams (if any).
2. Show detailed solution steps including all the calculations.
To determine if there will be a collision between the two cars, we need to analyze the forces at play during the braking process. Let's go through the solution step by step:
Step 1: Calculate the initial velocities of both cars:
- The car with passengers travels at a speed of 65 miles/hour. Convert this to meters per second (m/s): 65 miles/hour * 1609.34 meters/mile / 3600 seconds/hour = 29.057 m/s.
- The other car is traveling at 20 miles/hour. Convert this to m/s: 20 miles/hour * 1609.34 meters/mile / 3600 seconds/hour = 8.940 m/s.
Step 2: Determine the deceleration of the braking car:
- When the brakes are fully engaged and the tires are not rotating, the car decelerates with a constant negative acceleration equal to the friction between the tires and the road. Let's call this deceleration "a".
- The deceleration can be calculated using the equation: a = μ * g, where μ is the friction coefficient and g is the acceleration due to gravity (9.8 m/s^2).
Step 3: Calculate the time it takes for the braking car to stop:
- The time it takes for the car to stop can be calculated using the equation: t = (Vf - Vi) / a, where Vf is the final velocity (0 m/s) and Vi is the initial velocity of the car (29.057 m/s).
Step 4: Calculate the distance traveled by the braking car during the braking process:
- The distance traveled can be calculated using the equation: D = Vi * t + 0.5 * a * t^2.
Step 5: Compare the distance traveled by the braking car with the initial distance between the two cars:
- If the distance traveled by the braking car is less than the initial distance between the two cars (D = 30 meters), then there will be a collision. Otherwise, there will not be a collision.
Now, let's solve the problem using two different values for the friction coefficient (μ = 0.1 and μ = 0.8) and calculate the minimum safe distance for each case:
For μ = 0.1:
- Calculate the deceleration: a = 0.1 * 9.8 m/s^2 = 0.98 m/s^2.
- Calculate the time to stop: t = (0 - 29.057 m/s) / (-0.98 m/s^2) = 29.71 seconds.
- Calculate the distance traveled by the braking car: D = 29.057 m/s * 29.71 seconds + 0.5 * (-0.98 m/s^2) * (29.71 seconds)^2 = 247.58 meters.
- Since the distance traveled by the braking car (247.58 meters) is larger than the initial distance (30 meters), there will not be a collision.
For μ = 0.8:
- Calculate the deceleration: a = 0.8 * 9.8 m/s^2 = 7.84 m/s^2.
- Calculate the time to stop: t = (0 - 29.057 m/s) / (-7.84 m/s^2) = 3.70 seconds.
- Calculate the distance traveled by the braking car: D = 29.057 m/s * 3.70 seconds + 0.5 * (-7.84 m/s^2) * (3.70 seconds)^2 = 21.82 meters.
- Since the distance traveled by the braking car (21.82 meters) is smaller than the initial distance (30 meters), there will be a collision.
Therefore, the minimum safe distance between the two cars with the friction coefficient values of μ = 0.1 and μ = 0.8 would be 247.58 meters and 21.82 meters, respectively.