Question

Challenge A finch rides on the back of a Galapagos tortoise, which walks at the stately pace of \(0.060 \mathrm{~m} / \mathrm{s}\). After \(1.2\) minutes the finch tires of the tortoise's slow pace, and it takes flight, traveling in the same direction for another \(1.2\) minutes at \(13 \mathrm{~m} / \mathrm{s}\). What was the average speed of the finch over the entire \(2.4\)-minute interval?

Short answer

The average speed of the finch is approximately 6.53 m/s.

Step-by-step solution

Convert Time to Seconds

Both intervals of time are given in minutes. First, convert each interval to seconds. Since 1 minute equals 60 seconds:\[1.2 \text{ minutes} = 1.2 \times 60 = 72 \text{ seconds}\]The entire interval is:\[2.4 \text{ minutes} = 2.4 \times 60 = 144 \text{ seconds}\]

Calculate Distance Traveled by Tortoise

The tortoise moves at a speed of \(0.060\) m/s for \(72\) seconds. Calculate the distance using the formula \(\text{distance} = \text{speed} \times \text{time}\):\[\text{distance by tortoise} = 0.060 \times 72 = 4.32 \text{ meters}\]

Calculate Distance Traveled by Finch

The finch flies at a speed of \(13\) m/s for \(72\) seconds. Again, use the formula \(\text{distance} = \text{speed} \times \text{time}\):\[\text{distance by finch} = 13 \times 72 = 936 \text{ meters}\]

Calculate Total Distance Traveled

Add the distances traveled by the tortoise and the finch to find the total distance traveled by the finch:\[\text{total distance} = 4.32 + 936 = 940.32 \text{ meters}\]

Calculate Average Speed of the Finch

The average speed is given by the total distance divided by the total time. Use the total distance from Step 4 and the total time of 144 seconds:\[\text{average speed} = \frac{940.32}{144} \approx 6.53 \text{ m/s}\]

Key concepts

Unit Conversion

To solve physics problems effectively, mastering unit conversion is crucial. In this case, the problem provides time in minutes, which needs converting to seconds for calculations involving speed.

• One minute equals 60 seconds.
• Converting the finch and tortoise's travel times from minutes to seconds helps to keep units consistent.

For example, the problem gives the time as 1.2 minutes. Convert this by multiplying by 60 to get 72 seconds. Similarly, for 2.4 minutes, multiply by 60 to achieve 144 seconds. By ensuring all measurements are in compatible units, calculations become straightforward.

Distance Calculation

To find out how far both the tortoise and the finch traveled, use the distance formula, which is: \[ \text{distance} = \text{speed} \times \text{time} \]For the tortoise moving at 0.060 meters per second over a time of 72 seconds, multiply these values together to find the distance:

• 0.060 m/s \( \times \) 72 s = 4.32 meters

Likewise, for the finch flying at 13 meters per second for the same duration, calculate:

• 13 m/s \( \times \) 72 s = 936 meters

Summing these distances provides the total journey covered by the finch during its combined ride and flight.

Speed Formula

Understanding the speed formula is marked by its simplicity: speed is the rate of covering distance over time. This problem uses the formula to find the average speed, which is:\[ \text{average speed} = \frac{\text{total distance}}{\text{total time}} \]Where the finch's total distance traversed is 940.32 meters and the total time is 144 seconds, insert these values:

• \( \text{average speed} = \frac{940.32 \text{ m}}{144 \text{ s}} \approx 6.53 \text{ m/s} \)

The relationship outlined by the speed formula helps in understanding how speed reflects the changes in movement over time. This principle is applicable across various scenarios, whether on foot or wings.

Galapagos Finch and Tortoise Problem

The unique scenario of a finch hitching a ride on a tortoise, then flying, combines basic physics principles with a charming natural event. This problem explores how different modes of travel (slow-moving tortoise vs. fast-flying finch) influence the journey.

• Time spent: 1.2 minutes riding, 1.2 minutes flying.
• Speeds: 0.060 m/s for the tortoise, 13 m/s for the finch.

Combining these movements, the overall average speed takes into account both the leisurely pace of the tortoise and the rapid flight of the finch, showcasing how changing speeds affect average calculations.