Solving the Car Speed, Distance, and Time Problem

This problem involves two cars traveling the same distance but at different speeds and starting/finishing at different times. We need to find the total distance traveled, which is the distance between point A and point B.

Understanding the Problem Statement

Let's break down the information given:

• Car 1 starts from A to B at a speed of 20 km/h.
• Car 2 starts from A to B at a speed of 30 km/h.
• Car 2 starts \(1 \frac{1}{2}\) hours (1.5 hours) after Car 1.
• Car 2 reaches point B \(2 \frac{1}{2}\) hours (2.5 hours) before Car 1.

We need to find the distance between A and B.

Setting up the Equations

Let:

• \(D\) be the distance between point A and point B (in km).
• \(v_1\) be the speed of Car 1, so \(v_1 = 20\) km/h.
• \(v_2\) be the speed of Car 2, so \(v_2 = 30\) km/h.
• \(t_1\) be the time taken by Car 1 to travel from A to B (in hours).
• \(t_2\) be the time taken by Car 2 to travel from A to B (in hours).

The fundamental relationship between distance, speed, and time is:

\(\text{Distance} = \text{Speed} \times \text{Time}\)

From this, we can express time as:

\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)

Using this formula for each car:

• For Car 1: \(t_1 = \frac{D}{v_1} = \frac{D}{20}\)
• For Car 2: \(t_2 = \frac{D}{v_2} = \frac{D}{30}\)

Analyzing the Time Differences

The problem states that Car 2 starts 1.5 hours later than Car 1 and arrives 2.5 hours before Car 1. This means that the total travel time of Car 2 is significantly less than that of Car 1.

Consider the difference in travel times. Car 2 starts 1.5 hours later, so its journey is 1.5 hours shorter just by starting late. Additionally, it arrives 2.5 hours earlier, making its journey another 2.5 hours shorter.

Therefore, the total difference in travel time between Car 1 and Car 2 is the sum of the late start time and the early arrival time:

\(\text{Difference in time} = \text{Time Car 2 started later} + \text{Time Car 2 arrived earlier}\)

\(t_1 - t_2 = 1.5 \text{ hours} + 2.5 \text{ hours}\)

\(t_1 - t_2 = 4 \text{ hours}\)

So, Car 1 takes 4 hours longer than Car 2 to complete the journey.

Solving for the Distance (D)

Now we have an equation relating \(t_1\) and \(t_2\), and we have expressions for \(t_1\) and \(t_2\) in terms of \(D\).

Substitute the expressions for \(t_1\) and \(t_2\) into the equation \(t_1 - t_2 = 4\):

\(\frac{D}{20} - \frac{D}{30} = 4\)

To solve for \(D\), we need to combine the terms on the left side. Find the least common multiple (LCM) of 20 and 30, which is 60.

Multiply every term in the equation by 60 to eliminate the denominators:

\(60 \times \left(\frac{D}{20}\right) - 60 \times \left(\frac{D}{30}\right) = 60 \times 4\)

Simplify the terms:

\(3D - 2D = 240\)

\(D = 240\)

The distance between A and B is 240 km.

Verification

Let's check if this distance satisfies the conditions:

• Distance \(D = 240\) km.
• Time taken by Car 1: \(t_1 = \frac{D}{v_1} = \frac{240}{20} = 12\) hours.
• Time taken by Car 2: \(t_2 = \frac{D}{v_2} = \frac{240}{30} = 8\) hours.

Difference in travel times: \(t_1 - t_2 = 12 - 8 = 4\) hours.

This difference of 4 hours matches the condition derived from the start and arrival times (\(1.5 \text{ hours late start} + 2.5 \text{ hours early arrival} = 4 \text{ hours less travel time}\)).

The distance calculated is correct.

Summary of Calculations

Equation: \(\frac{D}{20} - \frac{D}{30} = 4\)

Solving gives \(D = 240\) km.

Revision Table: Car Speed and Distance

Additional Information: Speed, Distance, Time Problems

Speed, distance, and time problems are common in competitive exams. They often involve scenarios like:

• Constant speed travel
• Relative speed (objects moving towards or away from each other)
• Problems involving trains (considering lengths of trains and platforms)
• Problems involving boats and streams (upstream/downstream speed)
• Problems with variable speed or breaks during the journey

Key strategies for solving these problems include:

• Defining variables for unknown quantities (like distance or time).
• Using the basic formula \(D = v \times t\) and its variations.
• Setting up equations based on the given information, especially time differences or total time.
• Solving the resulting algebraic equations.
• Always checking the units (km, m, hours, seconds, km/h, m/s) for consistency.

Understanding how differences in start or end times affect the total travel duration is crucial for problems like the one discussed here.