In New York there is a long straight avenue through the city, there are three speed limit signs. THEY occur in the following order : 60, 30, and 15 km /h, with the 30 km/h sign being midway between the other two. Obeying these speed limits, the smallest possible time Oc that a driver can spend on this part of the road is to travel between the first and second signs at 60 km/h and between the second and third signs at 30 km/h. Another way of following the required speed limit is that a driver could slow down from 60 to 30 km/h with a constant deceleration and then do a similar thing from 30 to 15 km/h. This alternative requires a time Od. Find the ratio Od/Oc
To find the ratio Od/Oc, we need to calculate the time it takes to travel between the speed limit signs using both methods.
Method 1: Obeying the speed limits directly
The first sign has a speed limit of 60 km/h, so we assume the driver travels at this speed until reaching the second sign. The second sign has a speed limit of 30 km/h, so the driver slows down to this speed and continues until reaching the third sign, which has a speed limit of 15 km/h.
The time it takes to travel between the first and second signs at 60 km/h is given by the formula:
time1 = distance / speed = distance / 60
The time it takes to travel between the second and third signs at 30 km/h is given by the formula:
time2 = distance / speed = distance / 30
Since the second sign is midway between the first and third signs, we can assume the same distance is covered for both time1 and time2.
Method 2: Decelerating between speed limits
Here, we assume the driver decelerates from 60 km/h to 30 km/h with a constant deceleration and then similarly decelerates from 30 km/h to 15 km/h.
The formula to calculate the time taken to decelerate from one speed to another with a constant deceleration is given by:
time = (final speed - initial speed) / deceleration
Let's assume the deceleration is 'a' (constant for both speed changes).
The time it takes to decelerate from 60 km/h to 30 km/h is:
time1_dec = (30 - 60) / a
The time it takes to decelerate from 30 km/h to 15 km/h is:
time2_dec = (15 - 30) / a
Since the total time taken is the sum of the deceleration times, we can calculate:
Od = time1_dec + time2_dec = (30 - 60) / a + (15 - 30) / a = -30/a - 15/a = -45/a
To find the ratio Od/Oc, we need to calculate Od and Oc separately.
Now, let's calculate Oc using method 1:
Oc = time1 + time2 = distance / 60 + distance / 30 = 2 * distance / 60 = distance / 30
Comparing Od/Oc:
Od/Oc = (-45/a) / (distance / 30) = -45 / (a * distance)
We don't have enough information about the distance or the deceleration 'a' to compute the exact ratio Od/Oc.