Bridge Traffic Problem Overview

This problem requires us to calculate the maximum number of vehicles, including trucks and cars, that can efficiently use a single-lane bridge within a one-hour period. We must account for the specific lengths of each vehicle type, the mandatory safety gaps they need, and their constant travel speed.

Vehicle Dimensions and Gap Calculation

To determine the effective length segment each vehicle occupies on the bridge, we must combine its physical length with the minimum gap that must be maintained directly behind it. This combined length dictates how much space each vehicle 'uses' on the bridge.

• Truck Segment Length: A truck is 10 m long, and a gap of at least 20 m must be maintained after each truck. Therefore, the total effective length for one truck (including its required gap) is $\left(10 \text{ m} + 20 \text{ m}\right) = 30 \text{ m}$.
• Car Segment Length: A car is 5 m long, and a gap of at least 15 m must be maintained after each car. Therefore, the total effective length for one car (including its required gap) is $\left(5 \text{ m} + 15 \text{ m}\right) = 20 \text{ m}$.

Speed Unit Conversion for Bridge Traffic

The problem states that both trucks and cars travel at a constant speed of 36 km/h. To ensure consistency with the lengths provided in meters and to facilitate time calculations in seconds, we must convert this speed from kilometers per hour (km/h) to meters per second (m/s).

Speed $= 36 \text{ km/h}$

To convert km/h to m/s, we use the conversion factors: $1 \text{ km} = 1000 \text{ m}$ and $1 \text{ hour} = 3600 \text{ seconds}$.

Speed in m/s $= 36 \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}}$

Speed $= 36 \times \frac{1000}{3600} \text{ m/s}$

Speed $= 10 \text{ m/s}$.

Alternating Vehicle Flow Cycle Definition

The problem specifies that cars and trucks use the bridge alternately. This means the vehicles follow a repeating pattern: one truck, then one car, then one truck, and so on. We need to determine the total length of one complete cycle of this alternating flow.

• One complete cycle consists of one truck segment followed by one car segment.
• Length of one truck segment $= 30 \text{ m}$ (as calculated in 'Vehicle Dimensions and Gap Calculation').
• Length of one car segment $= 20 \text{ m}$ (as calculated in 'Vehicle Dimensions and Gap Calculation').
• Total length of one alternating cycle $= \text{Length of Truck Segment} + \text{Length of Car Segment}$
• Total length of one alternating cycle $= 30 \text{ m} + 20 \text{ m} = 50 \text{ m}$.

It's important to note that each such cycle includes exactly two vehicles: one truck and one car.

Cycle Time for Bridge Passage

Knowing the total length of one complete alternating cycle and the constant speed at which the vehicles travel, we can now calculate the exact time it takes for one full cycle to pass a specific point on the bridge.

The formula for time is given by: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

Time per cycle $= \frac{\text{Length of one alternating cycle}}{\text{Speed}}$

Time per cycle $= \frac{50 \text{ m}}{10 \text{ m/s}} = 5 \text{ seconds}$.

Hourly Vehicle Cycles on Bridge

To determine the maximum number of vehicles that can use the bridge in one hour, we first need to figure out how many of these complete alternating cycles can occur within that one-hour period.

Total time available in one hour $= 1 \text{ hour} = 60 \text{ minutes} \times 60 \text{ seconds/minute} = 3600 \text{ seconds}$.

Number of cycles in one hour $= \frac{\text{Total time available}}{\text{Time per cycle}}$

Number of cycles in one hour $= \frac{3600 \text{ seconds}}{5 \text{ seconds/cycle}} = 720 \text{ cycles}$.

Maximum Vehicles on Bridge Calculation

Each complete alternating cycle, as defined earlier, consists of one truck and one car, totaling two vehicles. Since we have calculated the total number of cycles that can pass in one hour, we can now determine the maximum total number of vehicles that can use the bridge within that time frame.

• Number of vehicles per cycle $= 2$ (1 truck + 1 car).
• Total number of cycles that can pass in one hour $= 720$.
• Maximum number of vehicles $= \text{Number of cycles} \times \text{Vehicles per cycle}$
• Maximum number of vehicles $= 720 \times 2 = 1440$.

Bridge Vehicle Capacity Summary

Below is a clear summary of the key parameters and the step-by-step results that lead to the final calculation of the maximum bridge vehicle capacity.