An athlete leaves one end of a pool of length L at
t = 0
and arrives at the other end at time t1. She swims back and arrives at the starting position at time t2. If she is swimming initially in the positive x direction, determine her average velocities symbolically in the first half of the swim, the second half of the swim, and the round trip. (Assume that time t2 is from the other end of the pool to the starting point. Use any variable or symbol stated above as necessary. Indicate the direction with the sign of your answer.)
(a) the first half of the swim
vavg =
(b) the second half of the swim
vavg =
(c) the round trip
vavg =
(d) What is her average speed for the round trip?
average speed =
Where do I even begin?
To determine the athlete's average velocities, we need to calculate the displacement and time for each segment of the swim.
Let's start by analyzing the first half of the swim. We know that the athlete starts at one end of the pool and arrives at the other end (a distance of L). The time it takes to complete this half of the swim is t1.
(a) The average velocity in the first half of the swim can be calculated using the formula:
vavg = displacement / time
The displacement in this case is L (since the athlete starts at one end and reaches the other) and the time is t1. So, the average velocity in the first half of the swim is:
vavg1 = L / t1 (positive direction since she is swimming initially in the positive x direction)
Next, let's consider the second half of the swim. In this case, the athlete swims back from the other end to the starting position (another distance of L). The time it takes to complete this half of the swim is t2.
(b) The average velocity in the second half of the swim can be calculated similarly:
vavg2 = displacement / time
Since the displacement is L (returning from the other end to the starting position) and the time is t2, the average velocity in the second half of the swim is:
vavg2 = L / t2 (negative direction since she is swimming back)
Finally, to find the average velocity for the round trip, we can add the average velocities for each half and divide by 2.
(c) The average velocity for the round trip is given by:
vavg_round_trip = (vavg1 + vavg2) / 2
Substituting the values from above:
vavg_round_trip = (L / t1 + L / t2) / 2
(d) Average speed is defined as the total distance traveled divided by the total time taken. In this case, the total distance traveled is 2L (since the athlete swims from one end to the other and then back) and the total time taken is t1 + t2.
So, the average speed for the round trip is:
average speed = total distance / total time
= 2L / (t1 + t2)
To summarize:
(a) Average velocity in the first half of the swim: vavg1 = L / t1 (positive direction)
(b) Average velocity in the second half of the swim: vavg2 = L / t2 (negative direction)
(c) Average velocity for the round trip: vavg_round_trip = (L / t1 + L / t2) / 2
(d) Average speed for the round trip: average speed = 2L / (t1 + t2)
Remember to substitute the given values of L, t1, and t2 to obtain specific numerical answers.