Calculating Average Speed for Bird Flight Path

To find the average speed, we need the total distance traveled and the total time taken. The bird flies along the three equal sides of an equilateral triangle. Let the length of each side be $s$ km.

Total Distance Calculation

The total distance covered is the sum of the lengths of the three sides: Total Distance = $s + s + s = 3s$ km.

Total Time Calculation

The time taken to cover each side depends on the speed during that part of the journey. We use the formula: Time = Distance / Speed.

• Time for the first side ($t_1$): $t_1 = \frac{s}{2}$ hours
• Time for the second side ($t_2$): $t_2 = \frac{s}{4}$ hours
• Time for the third side ($t_3$): $t_3 = \frac{s}{8}$ hours

The total time is the sum of the times for each side: Total Time = $t_1 + t_2 + t_3 = \frac{s}{2} + \frac{s}{4} + \frac{s}{8}$ hours.

To add these fractions, find a common denominator, which is 8: Total Time = $\frac{4s}{8} + \frac{2s}{8} + \frac{s}{8} = \frac{4s + 2s + s}{8} = \frac{7s}{8}$ hours.

Average Speed Calculation

Average speed is calculated as Total Distance divided by Total Time. Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}}$

Substitute the values we found: Average Speed = $\frac{3s}{\frac{7s}{8}}$ km/hr.

To simplify, multiply the numerator by the reciprocal of the denominator: Average Speed = $3s \times \frac{8}{7s}$ km/hr.

The $s$ variable cancels out: Average Speed = $\frac{3 \times 8}{7} = \frac{24}{7}$ km/hr.

Conclusion

The average speed of the bird is $\frac{24}{7}$ km/hr.