Problem 40 - Fundamentals of Physics

Question

You are driving toward a traffic signal when it turns yel- low. Your speed is the legal speed limit of \(v_{0}=55 \mathrm{~km} / \mathrm{h}\); your best deceleration rate has the magnitude \(a=5.18 \mathrm{~m} / \mathrm{s}^{2}\). Your best reaction time to begin braking is \(T=0.75 \mathrm{~s}\). To avoid having the front of your car enter the intersection after the light turns red, should you brake to a stop or continue to move at \(55 \mathrm{~km} / \mathrm{h}\) if the distance to the intersection and the duration of the yellow light are (a) \(40 \mathrm{~m}\) and \(2.8 \mathrm{~s}\), and (b) \(32 \mathrm{~m}\) and \(1.8 \mathrm{~s} ?\) Give an answer of brake, continuc, cither (if either strategy works), or neither (if neither strategy works and the yellow duration is inappropriate).

Step-by-Step Solution

Answer

(a) Brake, (b) Neither strategy works.

Step 1: Convert the Speed

First, convert the speed from km/h to m/s. The conversion factor is \(1 \text{ km/h} = \frac{1}{3.6} \text{ m/s}\). Thus, the initial speed \(v_0 = 55 \text{ km/h} = \frac{55}{3.6} \text{ m/s} \approx 15.28 \text{ m/s}\).

Step 2: Calculate Braking Distance

Use the formula for braking distance \(d = \frac{v_0^2}{2a}\), where \(v_0 = 15.28 \text{ m/s}\) and \(a = 5.18 \text{ m/s}^2\). Thus, \(d = \frac{(15.28)^2}{2 \times 5.18} \approx 22.52 \text{ m}\).

Step 3: Calculate Total Stopping Distance

Total stopping distance is the sum of the distance covered during the reaction time and the braking distance. The distance during reaction time \(d_{react} = v_0 \times T = 15.28 \times 0.75 \approx 11.46 \text{ m}\). The total stopping distance \(d_{total} = d_{react} + d = 11.46 + 22.52 \approx 33.98 \text{ m}\).

Step 4: Evaluate Case (a)

For (a), the intersection distance is \(40 \text{ m}\) and the yellow light lasts \(2.8 \text{ s}\). The distance covered at \(55 \text{ km/h}\) in \(2.8 \text{ s}\) is \(v_0 \times 2.8 = 15.28 \times 2.8 \approx 42.78 \text{ m}\). Since the total stopping distance is less than the intersection distance, braking will stop the car before the intersection. If continuing, the car will reach the intersection after light turns red. Decide to brake.

Step 5: Evaluate Case (b)

For (b), the intersection distance is \(32 \text{ m}\) and the yellow light lasts \(1.8 \text{ s}\). The distance covered at \(55 \text{ km/h}\) in \(1.8 \text{ s}\) is \(v_0 \times 1.8 = 15.28 \times 1.8 \approx 27.504 \text{ m}\). Here, both braking will stop the car before the intersection since stopping distance (33.98m) exceeds the intersection distance, neither choice clears the intersection in time. This suggests the yellow light duration is inappropriate; hence, neither option works effectively.

Key Concepts

Braking Distance

When you want to bring a moving vehicle to a stop, you need to consider your braking distance, which is the distance your car will travel from the time you hit the brakes until it comes to a complete stop. This is influenced by your speed and your car's deceleration ability. In this case, your initial speed needs to be converted to meters per second for easier calculation, as seen in the exercise: 55 km/h changes into approximately 15.28 m/s. The formula to compute braking distance is\[d = \frac{v_0^2}{2a}\]where \(v_0\) is the speed (15.28 m/s) and \(a\) is the deceleration rate (5.18 m/s²). By plugging in these values, we find out that the braking distance equals around 22.52 meters.Always remember:- Higher speeds require longer braking distances.- Slippery road conditions can increase braking distance.- The quality of brakes and tires play a crucial role as well.

Reaction Time

Reaction time is the period between noticing something (like a traffic signal turning red) and starting to brake. For this scenario, your reaction time is 0.75 seconds. During this time, your car will keep moving at its initial speed, covering some distance before you even begin to decelerate.To figure out how far your car travels during your reaction time, use the formula:\[d_{react} = v_0 \times T\]Here, \(v_0\) is 15.28 m/s, and \(T\) is 0.75 seconds, resulting in a reaction distance of about 11.46 meters. Thus, your total stopping distance is the sum of the reaction distance and the braking distance, which was calculated before to approximately 33.98 meters. Understanding your reaction time helps in planning your approach towards intersections safely.

Traffic Signal

Traffic signals guide the flow of traffic to ensure road safety. When approaching a traffic signal, it’s important to consider how much time you have to react and stop before the light turns red. The duration of the yellow light often indicates how soon you must decide to stop or proceed. For instance: - In scenario (a), with a yellow light duration of 2.8 seconds and a 40-meter distance to the intersection, you can brake safely to stop before the intersection as your stopping distance (33.98 meters) is sufficient. - In scenario (b), with a 1.8-second yellow light and a 32-meter distance, neither braking nor continuing safely crosses the intersection, highlighting that the yellow light duration may not provide enough time for safe decision-making. Always analyze if your stopping distance aligns with intersection distance and yellow light duration. This ensures you either cross safely or stop in time, avoiding potential accidents or fines for running a red light.