When the front of a 5.0 m long bus traveling at a constant speed of 20 m/s is 55 m away from an intersection, the traffic light turns yellow. It will take 3.0 s before the traffic light turns red. The driver’s reaction time is 0.25 s. The bus can accelerate up to 2.0 m/s2 or slow down at a maximum rate of 4.3 m/s2. The intersection is 22.0 m wide. Should the driver step on the brakes or on the gas pedal?

d = (V^2-Vo^2)/2a=(0-(20^2)/-8.6=46.51 m

= Stopping distance.

d = 55+22+5 = 82 m. = Distance through
intersection.

d = Vo*t + 0.5a*t^2 =
20*(3-0.25) + 0.5*2*(3-0.25)^2=62.6 m.

Therefore, the driver must step on the
brakes; because he or she cannot drive
the required 82 meters in 2.75 s.

Well, let's break this down, but don't worry, I won't break any traffic rules.

The bus is 55 m away from the intersection when the light turns yellow, and it will take 3.0 s for the light to turn red. The bus is traveling at a constant speed of 20 m/s and the driver's reaction time is 0.25 s.

So, in the 3.0 seconds, the bus will travel a distance of 20 m/s x 3.0 s = 60 m. Considering the bus is 5.0 m long, the front of the bus will be at 55 m + 60 m - 5.0 m = 110 m from the intersection when the light turns red.

The intersection is 22.0 m wide, which means that the bus would need to stop before reaching the 110 m mark. Since the driver's reaction time is 0.25 s, we need to consider this too.

If the driver steps on the brakes, the bus will start slowing down at its maximum deceleration rate of 4.3 m/s^2. We can use the equation v = u + at to determine how much distance the bus will travel while decelerating. The initial velocity (u) is 20 m/s, the acceleration (a) is -4.3 m/s^2 (negative because it's deceleration), and the final velocity (v) is 0 m/s.

Using the equation, we can calculate the distance as:

0 = 20 + (-4.3)t

-20 = -4.3t

t = 4.65 s

So, it will take approximately 4.65 s for the bus to come to a complete stop. During this time, the bus will travel a distance of:

s = ut + (1/2)at^2

s = 20(4.65) + (1/2)(-4.3)(4.65)^2

s = 93 + (-10)(21.6225)

s = 93 - 216.225

s = -123.225 m

Uh-oh! Negative distance? That's not possible! Seems like if the driver steps on the brakes, the bus won't be able to stop in time and will overshoot the intersection.

Therefore, it would be wiser for the driver to step on the gas pedal and accelerate the bus away from the intersection. We don't want to ruin anyone's day by crashing into the intersection, right?

To determine whether the driver should step on the brakes or the gas pedal, we need to calculate whether the bus will be able to stop before reaching the intersection within the time it takes for the traffic light to turn red.

Let's break down the problem step-by-step:

1. Calculate the distance the bus will travel during the driver's reaction time:
Distance = Speed x Time
Distance = 20 m/s x 0.25 s
Distance = 5 m

2. Calculate the distance the bus will have traveled until the traffic light turns red:
Distance = Speed x Time
Distance = 20 m/s x 3.0 s
Distance = 60 m

3. Calculate the total distance available for the bus to stop:
Total distance = Distance to intersection + Intersection width
Total distance = 55 m + 22 m
Total distance = 77 m

4. Calculate the maximum distance the bus can cover during the driver's reaction time:
Max distance during reaction time = Reaction time x (Initial Speed + Final Speed)/2
Max distance during reaction time = 0.25 s x (20 m/s + 0 m/s)/2
Max distance during reaction time = 0.25 s x 10 m/s
Max distance during reaction time = 2.5 m

5. Calculate the total distance the bus can cover before the traffic light turns red:
Total distance before red light = Total distance - Max distance during reaction time
Total distance before red light = 77 m - 2.5 m
Total distance before red light = 74.5 m

6. Calculate the distance the bus can cover while decelerating:
Distance covered while decelerating = (Final Speed^2 - Initial Speed^2) / (2 x Maximum Deceleration)
Distance covered while decelerating = (0 m/s^2 - (20 m/s)^2) / (2 x (-4.3 m/s^2))
Distance covered while decelerating = (-400 m^2/s^2) / (-8.6 m/s^2)
Distance covered while decelerating = 46.51 m

7. Determine if the bus can stop before the intersection:
If Distance covered while decelerating + Max distance during reaction time >= Total distance before red light, then the driver should step on the brakes. Otherwise, the driver should step on the gas pedal.

46.51 m + 2.5 m >= 74.5 m
49.01 m >= 74.5 m

Since 49.01 m is less than 74.5 m, the bus will not be able to stop before the intersection. Therefore, the driver should step on the brake pedal.

To determine whether the driver should step on the brakes or on the gas pedal, we need to calculate whether the bus can cross the intersection before the light turns red.

Let's break down the problem and calculate the time it takes for the bus to reach the intersection and the time it takes for the bus to cross the intersection once it starts decelerating.

1. Calculate the time it takes for the bus to reach the intersection:
The distance from the front of the bus to the intersection is 55 m.
The speed of the bus is 20 m/s.
Using the formula: time = distance / speed
time = 55 m / 20 m/s = 2.75 s
So, it takes 2.75 seconds for the bus to reach the intersection.

2. Calculate the time it takes for the bus to cross the intersection:
The width of the intersection is 22 m.
The bus needs to cross this distance while slowing down. Let's consider two scenarios:
- Scenario 1: The bus slows down at a maximum rate of 4.3 m/s²
- Scenario 2: The bus accelerates at a maximum rate of 2.0 m/s²

Scenario 1 - Deceleration:
To calculate the time it takes for the bus to stop at a deceleration rate of 4.3 m/s², we can use the following formula: time = (final velocity - initial velocity) / acceleration
The initial velocity is 20 m/s, the final velocity is 0 m/s, and the acceleration is -4.3 m/s² (negative because it's deceleration):
time = (0 m/s - 20 m/s) / -4.3 m/s² ≈ 4.65 s
So, in Scenario 1, it takes approximately 4.65 seconds for the bus to stop.

Scenario 2 - Acceleration:
To calculate the time it takes for the bus to cross the intersection while accelerating at 2.0 m/s², we need to calculate the distance traveled during acceleration and the time it takes to cover that distance.
First, let's calculate the distance traveled while accelerating:
Using the formula: distance = (final velocity² - initial velocity² ) / (2 * acceleration)
distance = (0 m/s² - 20 m/s)² / (2 * 2.0 m/s²) ≈ 200 m
So, the distance traveled during acceleration is approximately 200 meters.
Now, let's calculate the time it takes to cover that distance:
time = distance / speed
time = 200 m / 20 m/s = 10 s
Therefore, in Scenario 2, it takes 10 seconds for the bus to cross the intersection while accelerating.

3. Calculate the remaining time for the bus to stop after crossing the intersection:
The traffic light turns yellow when the bus is 55 m away from the intersection, and it will take 3 seconds for the light to turn red.
Given the driver's reaction time of 0.25 s, we subtract this time from the remaining time for the light to turn red:
remaining time = 3.0 s - 0.25 s = 2.75 s

Now, let's compare the times in both scenarios to the remaining time:

Scenario 1:
It takes 2.75 seconds for the bus to reach the intersection and approximately 4.65 seconds for the bus to stop.
Total time = 2.75 s + 4.65 s = 7.4 s

Scenario 2:
It takes 2.75 seconds for the bus to reach the intersection and 10 seconds for the bus to cross the intersection while accelerating.
Total time = 2.75 s + 10 s = 12.75 s

Since both scenarios take longer than the remaining time of 2.75 seconds before the light turns red, the driver should step on the brakes to stop the bus.