If a child ran into the road 60 to 65 feet ahead of your vehicle, what is the highest speed from which you could stop with good brakes before hitting him?
A. 50 mph
B. 40 mph
C. 20 mph
D. 30 mph
Answer (2)
Convert the distances to miles and assume a deceleration rate.
Use the kinematic equation v i = − 2 a d to calculate initial velocity.
Consider reaction time and the distance traveled during that time.
Based on the calculations and reaction time, determine the highest safe speed: 20 m p h .
Explanation
Problem Analysis Let's analyze the problem. We need to determine the highest speed at which a car can stop before hitting a child who is 60 to 65 feet away. We are given four possible speeds: 50 mph, 40 mph, 20 mph, and 30 mph. We need to select the best answer from these choices.
Distance Conversion First, let's convert the distances from feet to miles. There are 5280 feet in a mile. So, 60 feet is 5280 60 miles and 65 feet is 5280 65 miles.
Deceleration Rate Calculation Now, let's assume a reasonable deceleration rate for a car with good brakes. A typical deceleration rate is around 0.7g, where g is the acceleration due to gravity (32.2 ft/s^2). So, the deceleration rate is 0.7 × 32.2 = 22.54 ft/s^2. We need to convert this to miles per hour squared. 1 ft/s^2 is equal to 5280 360 0 2 miles/hour^2 which is approximately 24447.27 mph^2. Therefore, 22.54 ft/s^2 is equal to 22.54 × 5280 360 0 2 ≈ 550000 mph^2.
Kinematic Equation We will use the following kinematic equation: v f 2 = v i 2 + 2 a d , where v f is the final velocity (0 mph), v i is the initial velocity (the speed we want to find), 'a' is the deceleration (negative value), and 'd' is the stopping distance. Rearranging the equation to solve for v i , we get: v i = − 2 a d .
Initial Velocity Calculation Let's calculate v i for both distances (60 feet and 65 feet). For 60 feet, d = 5280 60 miles. So, v i = − 2 × ( − 550000 ) × 5280 60 = 5280 2 × 550000 × 60 = 12500 ≈ 111.8 mph. For 65 feet, d = 5280 65 miles. So, v i = − 2 × ( − 550000 ) × 5280 65 = 5280 2 × 550000 × 65 = 13541.67 ≈ 116.4 mph.
Reaction Time Consideration Now, we need to consider reaction time. Let's assume a reaction time of 1.5 seconds. During this time, the car travels at a constant speed before the brakes are applied. At 50 mph, the distance traveled in 1.5 seconds is 50 × 3600 1.5 × 5280 = 110 feet. This is more than the given distances, so 50 mph is not a possible answer. At 40 mph, the distance traveled in 1.5 seconds is 40 × 3600 1.5 × 5280 = 88 feet. This is also more than the given distances, so 40 mph is not a possible answer. At 30 mph, the distance traveled in 1.5 seconds is 30 × 3600 1.5 × 5280 = 66 feet. This is approximately equal to the larger distance, so 30 mph might be a possible answer. At 20 mph, the distance traveled in 1.5 seconds is 20 × 3600 1.5 × 5280 = 44 feet.
Refining the Approach Since the stopping distances we calculated (111.8 mph and 116.4 mph) are much higher than the given options, we need to refine our approach. The problem states that we need to select the one BEST answer. Given the reaction time consideration, the car traveling at 30 mph would travel 66 feet during the reaction time alone. This means that the car would hit the child before the driver even applies the brakes. Therefore, 30 mph is not a safe speed. The next lower speed is 20 mph. At 20 mph, the car travels 44 feet during the reaction time. This leaves 60 - 44 = 16 feet or 65 - 44 = 21 feet for braking.
Final Calculation and Conclusion Let's recalculate the required deceleration for 20 mph to stop within 16 to 21 feet. 20 mph is equal to 20 × 3600 5280 = 29.33 ft/s. Using the equation v f 2 = v i 2 + 2 a d , we have 0 = 29.3 3 2 + 2 a d . So, a = − 2 d 29.3 3 2 . For d = 16 feet, a = − 2 × 16 29.3 3 2 = − 26.85 ft/s^2. For d = 21 feet, a = − 2 × 21 29.3 3 2 = − 20.44 ft/s^2. These deceleration rates are plausible for a car with good brakes. Therefore, 20 mph is the most reasonable answer.
Examples
Understanding stopping distances is crucial for road safety. For example, knowing the relationship between speed, braking distance, and reaction time can help drivers maintain a safe following distance. If a driver knows that at 30 mph, their car travels 66 feet during their reaction time alone, they can make informed decisions about how much space to leave between their vehicle and the car in front of them. This knowledge is also essential for understanding how road conditions like rain or ice can affect stopping distances, potentially increasing the risk of accidents. By applying these concepts, drivers can significantly reduce the likelihood of collisions and ensure safer roads for everyone.
The highest speed from which you could stop with good brakes before hitting a child 60 to 65 feet away is 20 mph. This is concluded after calculating both reaction and stopping distances, confirming that higher speeds would not allow adequate stopping distance. Therefore, the correct answer is D. 20 mph .
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