Inge is competing in a 400 m race. She runs at a constant speed of 4.6 m/s for the first 150 m, then 4.2 m/s for 40 seconds. The remainder of the race takes her 38 seconds to complete. What is Inge’s average speed for the entire race to 1 dp?
The Correct Answer and Explanation is:
To find Inge’s average speed for the entire race, we first need to calculate the total distance she ran and the total time she took to complete the race.
Step 1: Calculate the total distance
The race is a 400-meter race, so the total distance covered is:
[
\text{Total distance} = 400 \, \text{m}
]
Step 2: Calculate the time taken for the first part of the race
Inge runs the first 150 meters at a constant speed of 4.6 m/s. To find the time taken for this part, we use the formula:
[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
]
[
\text{Time}_1 = \frac{150 \, \text{m}}{4.6 \, \text{m/s}} = 32.61 \, \text{seconds}
]
Step 3: Calculate the distance covered in the second part of the race
Inge runs at a speed of 4.2 m/s for 40 seconds. The distance covered during this time is:
[
\text{Distance}_2 = \text{Speed} \times \text{Time}
]
[
\text{Distance}_2 = 4.2 \, \text{m/s} \times 40 \, \text{s} = 168 \, \text{m}
]
Step 4: Calculate the remaining distance
After the first two parts of the race, Inge has run:
[
\text{Distance}_1 + \text{Distance}_2 = 150 \, \text{m} + 168 \, \text{m} = 318 \, \text{m}
]
The remaining distance to complete the race is:
[
\text{Remaining distance} = 400 \, \text{m} – 318 \, \text{m} = 82 \, \text{m}
]
Step 5: Calculate the time taken for the final part of the race
The remaining 82 meters take 38 seconds to complete.
Step 6: Calculate the total time
Now, we can calculate the total time taken for the entire race. The total time is the sum of the times for each part:
[
\text{Total time} = \text{Time}_1 + \text{Time}_2 + \text{Time}_3
]
[
\text{Total time} = 32.61 \, \text{s} + 40 \, \text{s} + 38 \, \text{s} = 110.61 \, \text{seconds}
]
Step 7: Calculate the average speed
The average speed is given by the formula:
[
\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}
]
[
\text{Average speed} = \frac{400 \, \text{m}}{110.61 \, \text{s}} \approx 3.6 \, \text{m/s}
]
Thus, Inge’s average speed for the entire race is 3.6 m/s (to 1 decimal place).