Concept explainers
(a)
The distance traveled by the vehicle when the acceleration is 45 ft/sec.
Answer to Problem 9P
x=443 ft
Explanation of Solution
Given information:
dudt=3.6−0.06uα=3.6β=0.06
Initial vehicle speed, u0=30 ft/sec
Final vehicle speed, ut=45 ft/sec
Concept used:
The time required by the vehicle upon acceleration is calculated by using the relation,
−βt=ln[(α−βut)(α−βu0)]t=(−1β)ln[(α−βut)(α−βu0)]
The distance travelled at any time is calculated by using the formula,
x=(αβ)t−αβ2(1−e−βt)+u0β(1−e−βt)
Calculation:
The time taken by the vehicle is calculated as,
t=(−1β)ln[(α−βut)(α−βu0)]=(−10.06)ln[3.6−0.06(45)3.6−0.06(30)]=11.55 sec
x=(αβ)t−αβ2(1−e−βt)+u0β(1−e−βt)=(3.60.06)11.55−3.6(0.06)2(1−e−(0.06×11.55))+300.06(1−e−(0.06×11.55))=693−499.93+249.96=443 ft
Conclusion:
The distance traveled when the vehicle has accelerated to 45 ft/sec is 443 ft.
(b)
The time taken by the vehicle to attain a speed of 45 ft/sec.
Answer to Problem 9P
t=11.55 sec
Explanation of Solution
Given information:
dudt=3.6−0.06uα=3.6β=0.06
Initial vehicle speed, u0=30 ft/sec
Final vehicle speed, ut=45 ft/sec
Concept used:
The time required by the vehicle after accelerated is calculated by using the relation,
−βt=ln[(α−βut)(α−βu0)]t=(−1β)ln[(α−βut)(α−βu0)]
Calculation:
t=(−1β)ln[(α−βut)(α−βu0)]=(−10.06)ln[3.6−0.06(45)3.6−0.06(30)]=11.55 sec
Conclusion:
The time for vehicle to attain speed of 45 ft/sec is 11.55 sec.
(c)
The acceleration of the vehicle after 4 seconds.
Answer to Problem 9P
a=1.42 fts2
Explanation of Solution
Given information:
dudt=3.6−0.06uα=3.6β=0.06
Initial vehicle speed, u0=30 ft/sec
Final vehicle speed, ut=45 ft/sec
Time taken, t=4 sec
Concept used:
The initial velocity is calculated and then the acceleration is calculated.
ut=αβ(1−e−βt)+u0e−βt
a=dudt
Calculation:
ut=αβ(1−e−βt)+u0e−βt=3.60.06(1−e−(0.06×4))+30(e−(0.06×4))=3.60.06(1−0.78663)+30(0.78663)=36.40 ft/sec
a=dudt=3.6−0.06u=3.6−0.06(36.40)=1.42 fts2
Conclusion:
The acceleration after 4 seconds is 1.42 fts2.
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Chapter 3 Solutions
MindTap Engineering for Garber/Hoel's Traffic and Highway Engineering, 5th Edition, [Instant Access], 1 term (6 months)
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