The "reaction time" of the average automobile driver is about 0.700 s. (The reaction time is the interval between the perception of a signal to stop and the application of the brakes.) If an automobile can slow down with an acceleration of 12.0 ft/s^{2}, compute the total distance covered in coming to a stop after a signal is observed (a) from an initial velocity of 15.0 mi/h (in a school zone) and (b) from an initial velocity of 55.0 mi/h.

The total stopping distance is:

X = Vo*(reaction time) + (a/2)*(deceleration time)^2

Reaction time = tr = 0.70 s
Deceleration time = Vo/a

X = Vo*tr + (1/2)(Vo^2)/a

(a) If Vo = 15 mi/h = 22 ft/s,
X = 22*(0.700) + (1/2)(22/15)^2/0.12
= 15.4 + 6.1 = 21.5 ft

(b) repeat calculation for Vo = 55 mi/h = 80.67 ft/s

To solve this problem, we need to break it down into several steps. Let's start by converting the initial velocities from miles per hour (mi/h) to feet per second (ft/s).

Step 1: Convert initial velocities to ft/s
To convert miles per hour to feet per second, we use the conversion factor 1 mi/h = 1.47 ft/s.

(a) Initial velocity = 15.0 mi/h
= 15.0 mi/h × 1.47 ft/s
= 22.05 ft/s

(b) Initial velocity = 55.0 mi/h
= 55.0 mi/h × 1.47 ft/s
= 80.85 ft/s

Now that we have the initial velocities in ft/s, we can proceed to the next step.

Step 2: Calculate the time taken to react (reaction time)
Given reaction time = 0.700 s

Step 3: Calculate the distance covered during the reaction time
To calculate the distance covered during the reaction time, we use the formula:
Distance = (1/2) × acceleration × (time)^2

Given acceleration = 12.0 ft/s^2
Reaction time = 0.700 s

Distance covered during reaction time = (1/2) × 12.0 ft/s^2 × (0.700 s)^2

Step 4: Calculate the distance covered to come to a stop
To calculate the distance covered to come to a stop, we consider two parts: distance covered during the reaction time and distance covered after the reaction time.

(a) Distance covered from initial velocity of 15.0 mi/h
Total distance = Distance covered during reaction time + Distance covered after reaction time

Distance covered after reaction time:
Using the formula:
Distance = (initial velocity)^2 / (2 × acceleration)
Distance covered after reaction time = (22.05 ft/s)^2 / (2 × 12.0 ft/s^2)

(b) Distance covered from initial velocity of 55.0 mi/h
Total distance = Distance covered during reaction time + Distance covered after reaction time

Distance covered after reaction time:
Using the formula:
Distance = (initial velocity)^2 / (2 × acceleration)
Distance covered after reaction time = (80.85 ft/s)^2 / (2 × 12.0 ft/s^2)

Once you plug in the values and perform the calculations, you will have the total distances covered in both scenarios.

To compute the total distance covered in coming to a stop after observing a signal, we need to consider the reaction time, acceleration, and initial velocity. Let's calculate it step by step:

Step 1: Convert the given initial velocities to ft/s.

(a) Initial velocity = 15.0 mi/h
1 mile = 5280 feet
1 hour = 3600 seconds

Converting the initial velocity to ft/s:
15.0 mi/h * (5280 ft/1 mi) * (1 hr/3600 s) = 22 ft/s

(b) Initial velocity = 55.0 mi/h

Converting the initial velocity to ft/s:
55.0 mi/h * (5280 ft/1 mi) * (1 hr/3600 s) = 80.67 ft/s

Step 2: Calculate the distance covered during the reaction time.

The distance covered during the reaction time can be calculated using the formula:
Distance = Velocity * Time

(a) Reaction time = 0.700 s
Distance (during reaction time) = 22 ft/s * 0.700 s = 15.4 ft

(b) Reaction time = 0.700 s
Distance (during reaction time) = 80.67 ft/s * 0.700 s = 56.47 ft

Step 3: Calculate the distance covered during deceleration.

The distance covered during deceleration can be calculated using the formula:
Distance = (Initial velocity^2) / (2 * acceleration)

(a) Initial velocity = 22 ft/s
Acceleration = -12.0 ft/s^2 (negative sign indicates deceleration)

Distance (during deceleration) = (22^2) / (2 * -12.0) = 10.17 ft

(b) Initial velocity = 80.67 ft/s
Acceleration = -12.0 ft/s^2

Distance (during deceleration) = (80.67^2) / (2 * -12.0) = 349.41 ft

Step 4: Calculate the total distance covered.

The total distance covered is the sum of the distances covered during the reaction time and deceleration.

(a) Total distance = Distance (during reaction time) + Distance (during deceleration)
Total distance = 15.4 ft + 10.17 ft = 25.57 ft

(b) Total distance = Distance (during reaction time) + Distance (during deceleration)
Total distance = 56.47 ft + 349.41 ft = 405.88 ft

So, the total distance covered in coming to a stop after observing a signal is 25.57 ft when the initial velocity is 15.0 mi/h and 405.88 ft when the initial velocity is 55.0 mi/h.