Calculating Total Distance with Changing Speed

This problem involves calculating the total distance Ajay travelled, given his initial speed, a change in speed mid-way, and the total time taken for the journey. We need to use the fundamental relationship between distance, speed, and time: Distance = Speed × Time, which can also be rearranged as Time = Distance / Speed.

Understanding the Journey Segments

The journey is divided into two equal halves in terms of distance:

• The first half of the distance is covered at a speed of 4 km/hr.
• The second half of the distance is covered at a speed that is double the initial speed, i.e., $2 \times 4$ km/hr = 8 km/hr.

The total time for the entire journey is 12 hours.

Setting Up the Calculation

Let's denote the total distance travelled by Ajay as $D$ kilometers.

Since he doubles his speed after reaching exactly half way, the distance of the first half is $\frac{D}{2}$ km, and the distance of the second half is also $\frac{D}{2}$ km.

Time Taken for Each Half

Using the formula Time = Distance / Speed:

• Time taken for the first half ($t_1$) = $\frac{\text{Distance of first half}}{\text{Speed in first half}} = \frac{D/2}{4}$ hours.
• Time taken for the second half ($t_2$) = $\frac{\text{Distance of second half}}{\text{Speed in second half}} = \frac{D/2}{8}$ hours.

Total Time Equation

The total time for the journey is the sum of the time taken for the first half and the second half. We are given that the total time is 12 hours.

So, $t_1 + t_2 = 12$ hours.

Substituting the expressions for $t_1$ and $t_2$:

$\frac{D/2}{4} + \frac{D/2}{8} = 12$

Solving for Total Distance D

Let's simplify the equation and solve for $D$:

$\frac{D}{8} + \frac{D}{16} = 12$

To add the fractions on the left side, we find a common denominator, which is 16.

Multiply the first term ($\frac{D}{8}$) by $\frac{2}{2}$:

$\frac{D}{8} \times \frac{2}{2} = \frac{2D}{16}$

Now, substitute this back into the equation:

$\frac{2D}{16} + \frac{D}{16} = 12$

Combine the terms on the left side:

$\frac{2D + D}{16} = 12$

$\frac{3D}{16} = 12$

To isolate $D$, multiply both sides by 16:

$3D = 12 \times 16$

$3D = 192$

Now, divide both sides by 3:

$D = \frac{192}{3}$

$D = 64$

So, the total distance travelled by Ajay is 64 kilometers.

Let's verify the times:

• Distance of first half = $D/2 = 64/2 = 32$ km. Speed = 4 km/hr. Time $t_1 = 32 / 4 = 8$ hours.
• Distance of second half = $D/2 = 64/2 = 32$ km. Speed = 8 km/hr. Time $t_2 = 32 / 8 = 4$ hours.

Total time = $t_1 + t_2 = 8 + 4 = 12$ hours, which matches the given total time. The calculation is correct.

Summary of Solution Steps

1. Identify the knowns: Initial speed (4 km/hr), total time (12 hours), speed change condition (doubles speed after half distance).
2. Identify the unknown: Total distance ($D$).
3. Define distances for each segment: First half is $D/2$, second half is $D/2$.
4. Calculate speed for each segment: 4 km/hr and 8 km/hr.
5. Express time for each segment using $D$: $t_1 = (D/2)/4$, $t_2 = (D/2)/8$.
6. Set up equation based on total time: $t_1 + t_2 = 12$.
7. Solve the equation for $D$.

Revision Table - Distance Speed Time Concepts

Additional Information - Speed, Distance, Time Problems

Speed, distance, and time problems are common in mathematics and physics. They often involve scenarios where speed is constant, or where speed changes over different parts of a journey, as seen in this question. Key to solving these problems is correctly identifying the known quantities and setting up an equation based on the relationship between distance, speed, and time for each part of the journey.

When the speed changes, you must calculate the time taken for each segment of the journey separately and then sum them up to get the total time, or sum the distances to get the total distance. Sometimes, problems might involve relative speed, such as when two objects are moving towards or away from each other.

Always ensure units are consistent (e.g., km and km/hr, or meters and m/s) before performing calculations.