Average Speed -4

How to analyse the problem: The particle travels equal distances along the three sides of an equilateral triangle, but its speeds are different on each side. Therefore, the time taken on each side will be different.

How to crack it: Find the time taken for each side using distance divided by speed. Then add the three distances and the three times. Finally, use total distance divided by total time.

Basic formulas:

For three sides:

Conclusion: Even when the distances are equal, average speed is not the simple average of the three speeds if the time intervals are different. Always use total distance divided by total time.

Step 1: Find the time taken along each side

Each side of the equilateral triangle is 30 m.

For the first side:

For the second side:

For the third side:

Step 2: Find the total distance

Step 3: Find the total time

Step 4: Find the average speed

How to analyse the problem: The particle travels along three sides of a right-angled triangle. The distances and speeds are different for the three sides, so the time taken on each side must be calculated separately.

How to crack it: Find the time taken for each side using distance divided by speed. Add all three distances and all three time intervals. Then divide total distance by total time.

Basic formulas:

Total distance:

Total time:

Conclusion: For motion along different sides of a triangle, average speed must be calculated using total distance divided by total time. The arithmetic average of the speeds cannot be used directly.

Step 1: Find the time taken along each side

For the 3 m side:

For the 4 m side:

For the 5 m side:

Step 2: Find the total distance

Step 3: Find the total time

Step 4: Find the average speed

How to analyse the problem: The journey contains two different types of paths. The particle first travels along the three sides of a triangle and then travels along a straight line. The total distance is the sum of all these path lengths.

How to crack it: Calculate the distance covered along each side of the triangle and add them. Then add the straight-line distance. Similarly, add all the time intervals. Finally, divide total distance by total time.

Basic formula:

Total distance for the complete journey:

For the three sides of the triangle:

Total time:

Conclusion: For a mixed journey involving a triangle and a straight path, calculate every actual path segment separately, add the distances and times, and then apply the average-speed formula.

Step 1: Find the distance along the triangle

The triangle is equilateral, so each side is 10 m.

Step 2: Add the straight-line distance

Step 3: Find the total time

The particle takes 5 s along each of the three sides and 5 s for the straight-line motion.

Step 4: Find the average speed

How to analyse the problem: The particle travels only along one side of the square. Therefore, the distance travelled is simply the length of that side.

How to crack it: Identify the length of the side as the total distance travelled. Then divide this distance by the time taken.

Basic formula:

If the particle travels one side of a square:

Therefore:

Conclusion: When a particle travels along only one side of a square, the distance travelled is equal to the side length. Average speed is obtained by dividing this distance by the time taken.

Step 1: Identify the distance travelled

The particle travels along one side of the square. The side length is 20 m.

Step 2: Identify the total time

Step 3: Find the average speed

How to analyse the problem: The journey is divided according to fractions of the total distance. Since the distances are given as fractions, the time taken for each part will be different.

How to crack it: Assume a convenient total distance. Divide it according to the given fractions. Find the time taken for each part using distance divided by speed. Then use total distance divided by total time.

Basic formulas:

If fractions f₁ and f₂ of the total distance are travelled at speeds v₁ and v₂:

The corresponding times are:

Conclusion: When different fractions of the total distance are travelled at different speeds, average speed is not obtained by simply averaging the speeds. The time taken by each distance fraction must be considered.

Step 1: Assume the total distance

Take the total distance as 3d so that the given fractions become simple distances.

Step 2: Find the distance covered at each speed

One-third of the total distance is covered at 20 m/s:

The remaining two-thirds is covered at 40 m/s:

Step 3: Find the time for each part

Time for the first part:

Time for the second part:

Step 4: Find the total time

Step 5: Find the average speed

How to analyse the problem: The total distance is divided into three parts. The first part is given directly as a fraction of the total distance. The remaining distance is then divided into two smaller parts.

How to crack it: First convert the nested fractions into three fractions of the total distance. Then find the time taken for each part using distance divided by speed. Finally, divide the total distance by the total time.

Basic formulas:

If the first fraction is f₁ and the remaining fraction is divided into f₂ and f₃:

The total time is the sum of the times taken for all three parts:

Conclusion: When the remaining fraction of a journey is divided into smaller fractions, first convert every part into a fraction of the total distance. Then calculate each time separately and use total distance divided by total time.

Step 1: Assume the total distance

Take the total distance as 4d. This makes all three distance fractions simple.

Step 2: Divide the total distance into three parts

First, the car travels 1/2 of the total distance:

The remaining 1/2 is divided into two equal parts. Therefore, each part is 1/4 of the total distance:

Step 3: Find the time for each part

For the first part:

For the second part:

For the third part:

Step 4: Find the total time

Step 5: Find the average speed

How to analyse the problem: The total distance is divided into three parts. First, a fraction of the total distance is given directly. The remaining distance is then divided into two smaller parts.

How to crack it: First convert the nested fractions into fractions of the total distance. Then find the time taken for each part using distance divided by speed. Add all the times and divide the total distance by the total time.

Basic formulas:

If the remaining fraction is divided in the ratio 1:2, the two smaller parts are obtained by dividing the remaining distance accordingly.

Conclusion: In a nested-fraction distance problem, first convert every part into a fraction of the total distance. Then calculate the time taken for each part and use total distance divided by total time.

Step 1: Assume the total distance

Take the total distance as 12d.

Step 2: Find the three distance parts

The first part is 1/2 of the total distance:

The remaining distance is 1/2 of the total distance:

This remaining 6d is divided in the ratio 1:2. Therefore, it is divided into 2d and 4d.

Therefore, the three fractions of the total distance are:

Step 3: Find the time taken for each part

First part is travelled at 20 m/s:

Second part is travelled at 30 m/s:

Third part is travelled at 40 m/s:

Step 4: Find the total time

Step 5: Find the average speed

How to analyse the problem: Here, the fractions are given for time, not distance. So, we divide the total time into the given fractions and calculate the distance travelled in each time interval.

How to crack it: When different speeds are given for different fractions of the total time, use total distance ÷ total time. Do not use the harmonic-mean formula, because that applies to equal distances.

Basic formula:

Suppose the total time is T. If the object travels with speeds v₁ and v₂ for fractions f and (1 − f) of the total time:

The corresponding distances are:

Therefore, the average speed becomes:

Conclusion: When speeds are given 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 3T.

Therefore:

Step 2: Calculate the distances

During the first interval:

During the second interval:

Step 3: Find total distance

Step 4: Find average speed

How to analyse the problem: The total distance is divided into equal fractions. Therefore, the three distances are equal. However, the speeds are different, so the time taken for each part will be different.

How to crack it: Since the distances are equal, we cannot take the ordinary arithmetic average of the speeds. Calculate the time taken for each distance and then use total distance ÷ total time.

Basic formulas:

For equal distances travelled at speeds v₁, v₂, v₃, let each distance be d. Then:

Therefore:

Cancelling d gives:

Conclusion: When equal fractions of the total distance are travelled at different speeds, the average speed is obtained from the reciprocal of the speeds. For two equal-distance parts, this reduces to the harmonic mean.

Step 1: Assume the total distance

Let the total distance be 3d. Since the distance is divided into three equal fractions, each part is d.

Step 2: Calculate the time for each part

Step 3: Find total time

Step 4: Find average speed

How to analyse the problem: Here, the total distance is divided into different fractions. Therefore, the distances travelled at the different speeds are not equal. The time taken for each part must be calculated separately.

How to crack it: First express each distance as a fraction of the total distance. Find the remaining fraction, calculate the time for each part using time = distance/speed, and finally use total distance ÷ total time.

Basic formulas:

If fractions of the total distance are f₁, f₂, f₃, then:

If the total distance is D, the individual distances are:

Therefore, the corresponding times are:

Hence, the general formula is:

Conclusion: For different fractions of distance travelled at different speeds, the average speed is determined by the distance fractions weighted through their corresponding times. It is not generally the arithmetic or harmonic mean of the speeds.

Step 1: Find the remaining fraction

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

Let the total distance be 12d.

Therefore:

Step 3: Calculate the time for each part

Step 4: Calculate total time

Step 5: Calculate average speed