Annie is walking to school leisurely at a speed of 1.0 m/s for the first half of her trip. She then remembers that her physics teacher told her that she must average 2.5 m/s to make it to school on time. Realizing she might be late, she starts running. However, after a quick calculation, she stops running--she's going to be late anyways.
a) show why she can't make it on time
If she completed the second half of her distance in zero time, her average speed would be 2 m/s, which is not fast enough.
To understand why Annie can't make it on time, we need to analyze the time she spent walking leisurely and the time she would have needed to run.
Let's assume the total distance from Annie's starting point to the school is d. Since Annie is walking leisurely, for the first half of the trip, she covers half the distance, which is d/2.
To find the total time taken for the first half of the trip, we can use the formula:
Time = Distance / Speed
So, the time taken for the first half of the trip would be:
Time1 = (d/2) / 1.0 = d/2
Next, Annie realizes that to make it on time, she needs to average a speed of 2.5 m/s for the entire trip. Since she has already covered half the distance at a speed of 1.0 m/s, she needs to calculate the remaining distance (d/2) she still needs to cover.
The remaining distance is:
Distance remaining = d - (d/2) = d/2
Now, let's calculate the time Annie would need to cover the remaining distance at an average speed of 2.5 m/s.
Time2 = (d/2) / 2.5 = (d/2)/(5/2) = (d/2) * (2/5) = d/5
Therefore, the total time needed to complete the entire trip at an average speed of 2.5 m/s would be the sum of Time1 and Time2:
Total Time = Time1 + Time2 = d/2 + d/5
To make it on time, Annie needs to complete the entire trip within this time. However, based on the given information and calculation, if Annie stops running, the total time she needs will be greater than d/2 + d/5, and she will be late.
Hence, based on the calculations, Annie can't make it on time even if she starts running.