Along a straight road through town, there are three speed-limit signs. They occur in the following order: 66 , 31 , and 19 mi/h, with the 31 -mi/h sign located midway between the other two. Obeying these speed limits, the smallest possible time tA that a driver can spend on this part of the road is to travel between the first and second signs at 66 mi/h and between the second and third signs at 31 mi/h. More realistically, a driver could slow down from 66 to 31 mi/h with a constant deceleration and then do the same thing from 31 to 19 mi/h. This alternative requires a time tB. Find the ratio tB/tA
90.5x10^3 :)
90.5x10^3 doesn't sound reasonable since it's a ratio but i went ahead and tried it and it didn't work i believe you have to combine the equations of motion i'm just not sure how to do it!
To find the ratio tB/tA, we need to first calculate the time tA spent traveling between the first and second signs at 66 mi/h, and the time tB spent slowing down from 66 mi/h to 31 mi/h with a constant deceleration.
First, let's calculate tA:
The distance between the first and second signs is equal to the distance between the second and third signs since the second sign is located midway between the other two. Therefore, we can consider the entire distance as one segment.
Distance = Total distance between the signs = 66 + 31 = 97 miles
Speed = 66 mi/h
Using the formula: Time = Distance / Speed
tA = 97 miles / 66 mi/h ≈ 1.47 hours
Now, let's calculate tB:
To calculate tB, we need to determine the deceleration required to slow down from 66 mi/h to 31 mi/h.
Acceleration = (Final Velocity - Initial Velocity) / Time
Since we have the final velocity (31 mi/h) and initial velocity (66 mi/h), we can rearrange the formula to solve for time:
Time = (Final Velocity - Initial Velocity) / Acceleration
Acceleration = (31 mi/h - 66 mi/h) / tB
Next, we need to calculate the distance covered during the deceleration phase. We know that the time taken to cover this distance (tA) is the same as the time taken at a constant speed of 66 mi/h for the same distance. Therefore:
Distance = 97 miles
Next, using the formula:
Distance = Initial Velocity * Time + (1/2) * Acceleration * Time^2
97 = 66 * tB + (1/2) * Acceleration * tB^2
Now, substitute the value of acceleration from the previous step:
97 = 66 * tB + (1/2) * ((31 - 66) / tB) * tB^2
Simplifying the equation, we get:
97 = 66tB + (1/2) * (-1) * tB^2
Rearranging the equation:
(tB^2) - 132tB + 194 = 0
Solving this quadratic equation, we can find the value of tB.
Using the quadratic formula, where a = 1, b = -132, and c = 194:
tB = (-(-132) ± √((-132)^2 - 4 * 1 * 194)) / (2 * 1)
tB ≈ 122.34 or tB ≈ 9.666
Since we are looking for a realistic value for tB, we use tB ≈ 9.666 hours.
Finally, to find the ratio tB/tA:
tB/tA ≈ 9.666 / 1.47
tB/tA ≈ 6.583
Therefore, the ratio tB/tA is approximately 6.583.