What is the overtaking time if the speed of overtaking and overtaken vehicles are 72.5 and 31.5 km/h respectively? Assume average acceleration rate during overtaking to be 0.99 m/sq. s.
- A6.0 s
- B7.0 sCorrect
- C5.0 s
- D8.0 s
Solution
Concept
The actual overtaking operation time (T) depends on the minimum space headway between vehicles (s) and the average rate of acceleration (a) of the overtaking vehicle.
Formula
s = 0.2 Vb + 6 (where Vb is in km/h), T = √(4s / a)
Step-by-step solution
1. Calculate the minimum space headway s: s = 0.2 × 31.5 + 6 = 12.3 m 2. Calculate the time of overtaking maneuver T: T = √(4 × 12.3 / 0.99) = √49.697 ≈ 7.05 s ≈ 7.0 s
Explanation
The time taken during the actual overtaking maneuver is given by T = √(4s / a), where s is the minimum space headway between vehicles and a is the acceleration. For an overtaken vehicle speed of 31.5 km/h, the space headway is s = 0.2Vb + 6 = 0.2(31.5) + 6 = 12.3 m. Substituting s = 12.3 m and a = 0.99 m/s² yields T = √(4 × 12.3 / 0.99) ≈ 7.05 seconds, which rounds to 7.0 s.
Why the other options are wrong
A: Incorrect evaluation or using a wrong formula for spacing s. C: Result of neglecting the constant 6 in the spacing formula. D: Result of using the speed of the overtaking vehicle (72.5 km/h) instead of the overtaken vehicle (31.5 km/h).
Shortcut
Since 4s ≈ 4 × 12.3 = 49.2, and a ≈ 1, T ≈ √49 = 7 s.
Common mistake
Confusing speed in km/h with m/s when calculating spacing s = 0.7vb + 6, or using the total speed instead of the overtaken vehicle's speed.
Correct answer: B) 7.0 s