Average Speed -5

How to analyse the problem: The journey is divided into two parts. The first part is a given fraction of the total distance, and the second part is simply the remaining distance.

How to crack it: First find the fraction of the remaining distance. Then assume a convenient total distance and calculate the time taken for each part. Finally, use total distance ÷ total time.

Basic formulas:

If the first part is a fraction f of the total distance, then the remaining fraction is:

If the total distance is D, then:

For speeds v₁ and v₂, the times are:

Therefore, the derived formula is:

Conclusion: When a fraction of the journey is travelled at one speed and the remaining distance at another speed, use the distance fractions to determine the individual times. The average speed is generally not the arithmetic average of the two speeds.

Step 1: Find the remaining fraction

First part: 2/5 of the total distance.

Step 2: Assume the total distance

Let the total distance be 5d.

Therefore:

Step 3: Calculate the time taken

Step 4: Find total time

Step 5: Find average speed

How to analyse the problem: Here, the journey is divided according to time. A fraction of the total time is spent at one speed, while the remaining time is spent at another speed.

How to crack it: Since the time fractions are given directly, first find the time spent at each speed. Then calculate the distance covered in each interval and use total distance ÷ total time.

Basic formulas:

If the first part occupies a fraction f of the total time T, then:

If the corresponding speeds are v₁ and v₂, the distances are:

Therefore:

Conclusion: When different speeds are maintained for different fractions of the total time, the average speed is the time-weighted average of the speeds.

Step 1: Assume the total time

Let the total time be 5T.

The car travels at 20 m/s for 2/5 of the total time:

The remaining time is:

Step 2: Calculate the distance in each interval

Step 3: Find total distance

Step 4: Find average speed

How to analyse the problem: This is a mixed-fraction problem. One part of the journey is specified as a fraction of distance, while the remaining part is divided according to fractions of time. We must treat the distance fraction and time fractions separately.

How to crack it: First determine the distance covered in the part where a distance fraction is given. Then use the given speed to determine its time. For the remaining part, use the time fractions to determine the speeds and distances. Finally apply total distance ÷ total time.

Basic formulas:

For a known fraction f of total distance D:

Its time is:

If the remaining time is divided into fractions g and (1 − g), then:

Conclusion: In mixed problems, always identify whether each fraction refers to distance or time. Distance fractions determine distances, while time fractions determine times. Do not mix the two directly.

Step 1: Assume the total distance

Let the total distance be 3d.

The first part is 1/3 of the total distance:

Therefore, the remaining distance is:

Step 2: Find the time for the first part

The first part is travelled at 20 m/s.

Step 3: Analyse the remaining time

Let the remaining time be 2T. Since it is divided equally between the two speeds:

The distances travelled during these two intervals are:

Their total distance is the remaining distance 2d:

Therefore, the total time for the remaining distance is:

Step 4: Find total time

Step 5: Find average speed

How to analyse the problem: The journey contains two moving portions and a stopping/waiting interval. The waiting time does not add to the distance, but it does add to the total time.

How to crack it: First calculate the distance of each moving part from the given fractions. Find the time taken for each moving part and then add the waiting time to obtain the total time. Finally use total distance divided by total time.

Basic formulas:

If the object stops for a waiting time t_w:

For a journey divided into distance fractions:

The moving times are:

Therefore, including a stop:

Conclusion: A stop or waiting period always contributes to total time, even though it contributes zero distance. Therefore, it must be included when calculating average speed for the complete journey.

Step 1: Calculate the two distances

Total distance = 500 m. The first part is 2/5 of the total distance:

Therefore, the remaining distance is:

Step 2: Calculate the moving times

Time for the first part:

Time for the second part:

Step 3: Include the waiting time

Waiting time = 10 s.

Step 4: Calculate average speed

How to analyse the problem: The journey is divided into three fractional distances, and each fraction is travelled at a different speed. The three fractions together must account for the entire journey.

How to crack it: First find the fraction of the remaining distance. Then assume a convenient total distance and calculate the distance and time for each of the three parts. Finally use total distance ÷ total time.

Basic formula:

If the three distance fractions are f₁, f₂, f₃, then:

If the total distance is D, then:

The corresponding times are:

Therefore, for three fractional distances:

Conclusion: When a journey is divided into three different distance fractions and each part has a different speed, calculate the time of each part separately. The average speed is determined by the total distance and total time, not by simply averaging the three speeds.

Step 1: Find the remaining fraction

The given fractions are 1/4 and 1/3. Therefore, the remaining fraction is:

So the three distance fractions are 1/4, 1/3 and 5/12.

Step 2: Assume the total distance

Take the total distance as 12d.

Step 3: Calculate the time for each part

Step 4: Find total time

Taking the common denominator 60:

Step 5: Find average speed

How to analyse the problem: This is an advanced fractional problem because the journey is divided using a nested distance fraction, followed by a stop. The second fraction applies only to the remaining distance, not to the original total distance.

How to crack it: First determine every actual distance fraction. Then calculate the time for each moving section. Add the stopping time to the moving times. Finally calculate average speed using total distance divided by the complete journey time.

Basic formulas:

If the first part is a fraction f of the total distance D:

If a fraction g of the remaining distance is travelled in the second part:

The total time, including a stop of duration t_w, is:

Conclusion: In advanced fractional problems, always ask: “Fraction of what?” A fraction may refer to the total distance or only the remaining distance. Stops must then be added separately to the total time.

Step 1: Calculate the first distance

Total distance = 600 m. The first part is 1/4 of the total distance:

Step 2: Find the remaining distance

The second part is 1/2 of this remaining distance:

Therefore, the final distance is:

Step 3: Calculate the moving times

Step 4: Include the stopping time

The car stops for 10 s.

Step 5: Calculate average speed