Along a straight road through town, there are three speed-limit signs. They occur in the following order: 56 , 33 , and 24 mi/h, with the 33 -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 56 mi/h and between the second and third signs at 33 mi/h. More realistically, a driver could slow down from 56 to 33 mi/h with a constant deceleration and then do the same thing from 33 to 24 mi/h. This alternative requires a time tB. Find the ratio tB/tA
To find the ratio tB/tA, we need to calculate both tB and tA.
First, let's calculate tA, assuming the driver travels at the given speed limits:
For the first segment between the first and second sign:
Distance = difference in speed limits = 56 - 33 = 23 mi
Time = Distance/Speed = 23/56 hours
For the second segment between the second and third sign:
Distance = difference in speed limits = 33 - 24 = 9 mi
Time = Distance/Speed = 9/33 hours
Total time tA = time for the first segment + time for the second segment = 23/56 + 9/33 hours
Now, let's calculate tB, assuming the driver decelerates from 56 to 33 mi/h and then from 33 to 24 mi/h:
To decelerate from 56 to 33 mi/h:
Initial speed = 56 mi/h
Final speed = 33 mi/h
Deceleration (negative acceleration) = change in speed / time taken
Assuming the deceleration is constant, let's denote the time taken as t1.
33 - 56 = -23 mph (negative because it's a deceleration)
Deceleration = -23 mph / t1
To decelerate from 33 to 24 mi/h:
Initial speed = 33 mi/h
Final speed = 24 mi/h
Deceleration (negative acceleration) = change in speed / time taken
Assuming the deceleration is constant, let's denote the time taken as t2.
24 - 33 = -9 mph (negative because it's a deceleration)
Deceleration = -9 mph / t2
From calculus, we know that the time taken to decelerate is given by:
Time taken = (Final Velocity - Initial Velocity) / Acceleration
For t1:
Time taken = (33 - 56) / Deceleration = -23 mph / (-23 mph / t1) = t1
For t2:
Time taken = (24 - 33) / Deceleration = -9 mph / (-9 mph / t2) = t2
Total time tB = t1 + t2
Now, we can find the ratio tB/tA:
Ratio tB/tA = (t1 + t2) / (23/56 + 9/33) = (t1 + t2) / (1848/33 + 504/33)
= (t1 + t2) / (2352/33)
= 33(t1 + t2) / 2352
So, the ratio tB/tA is 33(t1 + t2) / 2352.
To find the ratio tB/tA, we need to calculate the times tA and tB.
First, let's calculate tA. The driver travels at 56 mi/h between the first and second signs, and then at 33 mi/h between the second and third signs. Since the second sign is located midway between the other two, the distance between the first and second signs is the same as the distance between the second and third signs.
Let's assume the distance between the first and second signs (and also the second and third signs) is d miles.
The time tA taken to travel between the first and second signs at 56 mi/h is given by:
tA = d/56 (since time = distance/speed)
The time tA taken to travel between the second and third signs at 33 mi/h is also d/33.
Therefore, the total time tA is:
tA = d/56 + d/33
Now, let's calculate tB. The driver needs to decelerate from 56 mi/h to 33 mi/h, and then from 33 mi/h to 24 mi/h.
To decelerate from 56 mi/h to 33 mi/h, let's assume the driver takes time t1.
Using the equations of motion, we can calculate the time t1:
33 = 56 - (acceleration * t1)
Similarly, to decelerate from 33 mi/h to 24 mi/h, let's assume the driver takes time t2.
Using the equations of motion, we can calculate the time t2:
24 = 33 - (acceleration * t2)
Now, using the distance formula, we can relate the distances covered during the decelerations:
d = 56 * t1 - (0.5 * acceleration * t1^2) + 33 * t2 - (0.5 * acceleration * t2^2)
Simplifying, we get:
d = 56 * t1 - 0.5 * acceleration * t1^2 + 33 * t2 - 0.5 * acceleration * t2^2
Since we know that the distance between the first and second signs (and also the second and third signs) is d, we can rewrite the above equation as:
d = 56 * t1 - 0.5 * acceleration * t1^2 + 33 * t2 - 0.5 * acceleration * t2^2 = d
Now, we have two equations with two unknowns (t1 and t2). We can solve these equations simultaneously to find t1 and t2.
Once we have t1 and t2, we can calculate tB:
tB = t1 + t2
Finally, we can find the ratio tB/tA:
tB/tA = (t1 + t2)/(d/56 + d/33)
Let L be the distance between signs. It is the same for the 56 and the 33 mph segments.
tA = L/56 + L/33 = L/20.764
tB = L/44.5 + L/28.5 = L/17.373
44.5 and 28.5 are the average speeds in the two segments, while decelerating.
tB/tA = 1.195