Along a straight road through town, there are three speed-limit signs. They occur in the following order: 51 , 33 , and 26 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 51 mi/h and between the second and third signs at 33 mi/h. More realistically, a driver could slow down from 51 to 33 mi/h with a constant deceleration and then do the same thing from 33 to 26 mi/h. This alternative requires a time tB. Find the ratio tB/tA
To find the ratio tB/tA, we need to calculate the times tA and tB.
First, let's find the time tA.
Given:
Speed between the first and second sign = 51 mi/h
Speed between the second and third sign = 33 mi/h
We can calculate tA using the formula:
tA = Distance / Speed
The distance between the first and second sign is the same as the distance between the second and third sign, as the second sign is located midway between the other two signs.
Let's say the distance between the signs is D.
So, tA = 2D / (51 + 33)
Now, let's find the time tB.
Given:
Initial speed = 51 mi/h
Final speed = 33 mi/h
Initial speed between the second and third sign = 33 mi/h
Final speed between the second and third sign = 26 mi/h
To find tB, we need to find the time it takes to decelerate from 51 mi/h to 33 mi/h and then from 33 mi/h to 26 mi/h.
Let's say the time it takes to decelerate from 51 mi/h to 33 mi/h is t1, and the time it takes to decelerate from 33 mi/h to 26 mi/h is t2.
Using the formulas of uniformly decelerated motion:
t1 = (Final Speed - Initial Speed) / Acceleration
t2 = (Final Speed - Initial Speed) / Acceleration
The deceleration is the same for both cases.
So, tB = t1 + t2
Now, let's calculate the values.
Given:
Initial Speed = 51 mi/h
Final Speed = 33 mi/h
Initial Speed between the second and third sign = 33 mi/h
Final Speed between the second and third sign = 26 mi/h
Acceleration can be calculated using the formula:
Acceleration = (Final Speed - Initial Speed) / Time
Let's find the acceleration.
Acceleration = (33 - 51) / t1
Acceleration = -18 / t1
Acceleration = (26 - 33) / t2
Acceleration = -7 / t2
Since the acceleration is the same in both cases, we can set -18 / t1 = -7 / t2 and solve for t1 in terms of t2.
-18 / t1 = -7 / t2
Cross-multiply:
18t2 = 7t1
t1 = (18/7) t2
Now, substitute the value of t1 in the equation tB = t1 + t2.
tB = (18/7) t2 + t2
tB = (25/7) t2
Now, we have the values of tA and tB. Let's find the ratio tB/tA.
Ratio tB/tA = tB / tA
Ratio tB/tA = (25/7) t2 / (2D / (51 + 33))
Ratio tB/tA = (25/7) t2 * (51 + 33) / (2D)
Simplifying further,
Ratio tB/tA = (25/7) * (51 + 33) / 2D
Ratio tB/tA = (25/7) * 84 / 2D
Ratio tB/tA = (25/7) * 42 / D
So, the ratio tB/tA is (25/7) * 42 / D.
To find the ratio tB/tA, we need to calculate the time it takes to travel between the speed limit signs using the given information.
For tA, the smallest possible time, the driver travels at 51 mi/h between the first and second signs and at 33 mi/h between the second and third signs. We can calculate the time for each segment using the formula time = distance / speed.
The distance between the first and second signs is the same as the distance between the second and third signs since the second sign is located midway between them.
Let's assume the distance between each sign is d miles.
Time to travel from first to second sign at 51 mi/h:
t1A = d / 51
Time to travel from second to third sign at 33 mi/h:
t2A = d / 33
The total time tA is the sum of t1A and t2A:
tA = t1A + t2A = d / 51 + d / 33
Now, let's calculate tB, where the driver slows down from 51 mi/h to 33 mi/h and then from 33 mi/h to 26 mi/h.
For tB, since we are considering a deceleration, we can use the formula for deceleration with constant acceleration:
v_f^2 = v_i^2 + 2ad
Where:
- v_f is the final velocity
- v_i is the initial velocity
- a is the deceleration
- d is the distance
To find the deceleration, we can rearrange the formula:
a = (v_f^2 - v_i^2) / (2d)
Let's calculate the deceleration from 51 mi/h to 33 mi/h:
a1 = (33^2 - 51^2) / (2d)
Similarly, let's calculate the deceleration from 33 mi/h to 26 mi/h:
a2 = (26^2 - 33^2) / (2d)
The time t1B it takes to decelerate from 51 mi/h to 33 mi/h can be calculated using the formula:
t1B = (v_f - v_i) / a1
Similarly, the time t2B it takes to decelerate from 33 mi/h to 26 mi/h can be calculated using the formula:
t2B = (v_f - v_i) / a2
The total time tB is the sum of t1B and t2B:
tB = t1B + t2B
Now, to find the ratio tB/tA, we can substitute the calculated values into the formula and simplify:
tB/tA = (t1B + t2B) / (t1A + t2A)
Substituting the expressions for t1A, t2A, t1B, and t2B:
tB/tA = [(v_f - v_i) / a1 + (v_f - v_i) / a2] / [d / 51 + d / 33]
Simplifying the expression further will give you the ratio tB/tA.