A person driving her car at 54 km/h approaches an intersection just as the traffic light turns yellow. She knows that the yellow light lasts only 2.0 s before turning to red, and she is 30 m away from the near side of the intersection. The intersection is 15 m wide. Her car's maximum deceleration is -6.6 m/s2, whereas it can accelerate from 54 km/h to 70 km/h in 4.8 s. Ignore the length of her car and her reaction time.
If she hits the brakes, how far will she travel before stopping?
If she hits the gas instead, how far will she travel before the light turns red?
Should she try to stop, or should she speed up to cross the intersection before the light turns red?
yes
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To find the distance traveled before stopping, we need to determine the deceleration and the time it takes to stop.
1. Firstly, we need to convert the speed from km/h to m/s. We know that 1 m/s is equal to 3.6 km/h, so the car's initial speed is (54 km/h) / (3.6 m/s) = 15 m/s.
2. To determine the deceleration, we use the formula: acceleration = (final velocity - initial velocity) / time. Here, the final velocity is 0 m/s (since she stops), the initial velocity is 15 m/s, and the time is the time it takes to stop.
3. Using the given maximum deceleration of -6.6 m/s², we can rearrange the formula for acceleration as: time = (final velocity - initial velocity) / acceleration = (0 m/s - 15 m/s) / -6.6 m/s².
4. By calculating, we find that the time it takes for the car to stop is approximately 2.27 s.
5. Now, to determine the distance traveled before stopping, we use the formula: distance = initial velocity * time + 0.5 * acceleration * time². Here, the initial velocity is 15 m/s, the time is 2.27 s, and the acceleration is -6.6 m/s².
6. By plugging in the values, we find that the distance traveled before stopping is approximately 17.69 meters.
Therefore, if she hits the brakes, she will travel approximately 17.69 meters before stopping.
To find the distance traveled before the light turns red if she hits the gas instead, we need to determine the time it takes for the light to turn red.
1. We are given that the yellow light lasts for 2.0 s before turning red.
2. The time it takes for the light to turn red is therefore 2.0 s.
To find out whether she should try to stop or speed up, we need to compare the distance traveled before stopping (17.69 m) with the distance she would travel if she sped up and crossed the intersection before the light turns red.
1. To determine this distance, we need to calculate the time it takes for the car to accelerate from 15 m/s to 70 km/h.
2. We can use the formula for acceleration: acceleration = (final velocity - initial velocity) / time. Here, the final velocity is 70 km/h, which is equivalent to (70 km/h) / (3.6 m/s) = 19.44 m/s. The initial velocity is 15 m/s, and the time is the time it takes to accelerate.
3. Rearranging the formula, we have: time = (final velocity - initial velocity) / acceleration = (19.44 m/s - 15 m/s) / acceleration.
4. By calculating, we find that the time it takes for the car to accelerate is approximately 4.4 s.
5. To find the distance traveled before the light turns red, we use the formula: distance = initial velocity * time + 0.5 * acceleration * time². Here, the initial velocity is 15 m/s, the time is 4.4 s, and the acceleration is the maximum acceleration of the car.
6. Plugging in the values, we find that the distance traveled before the light turns red is approximately 36.33 meters.
Therefore, if she hits the gas instead, she would travel approximately 36.33 meters before the light turns red.
Based on the distances calculated, she should try to speed up to cross the intersection before the light turns red. However, it is important to note that these calculations assume ideal conditions and do not account for factors such as road conditions, other vehicles, or the driver's reaction time. It is always recommended to prioritize safety and adhere to traffic regulations.