Question

An athlete leaves one end of a pool of length $\mathit{L}$ at $\mathit{t}\mathbf{=}\mathbf{0}$ and arrives at the other end at time ${\mathit{t}}_{\mathbf{1}}$.She swims back and arrives at the starting position at time ${\mathit{t}}_{\mathbf{2}}$. If she is swimming initially in the positive $\mathit{x}$ direction, determine her average velocities symbolically in

(a) the first half of the swim,

(b) the second half of the swim, and

(c) the round trip.

(d) What is her average speed for the round trip?

Short answer

a) Her average velocities symbolically in the first half of the swim is $\frac{L}{t_1}$.

b) Her average velocities symbolically in the second half of the swim is$-\frac{L}{t_2}$.

c) Her average velocities symbolically the round trip is zero .

d) Her average speed for the round trip is $\frac{2L}{t_1+t_2}$.

Step-by-step solution

Definition of average velocity.

Average velocity is the ratio of the overall distance travelled in a certain direction to the total amount of time it took the body to travel that distance.The average velocity of a particle is,

$$\begin{gathered} v_{\text{avg}}=\frac{\Delta x}{\Delta t} \\ =\frac{x_f-x_i}{t_f-t_i} \end{gathered}$$

Here,$\Delta x$represents the particle's displacement along the $x$axis, $\Delta t$represents the time interval, $x_f$represents the particle's final position, $x_i$represents the particle's initial position, $t_f$represents the time observed at the final position, and$t_i$ represents the time observed at the initial position.

(a) Determining her average velocities in the first half of the swim.

The rate of change of displacement with respect to time is defined as the person's average velocity.

That is,

$$\begin{gathered} v_{avg}=\frac{\Delta x}{\Delta t} \\ =\frac{L-0}{t_1} \end{gathered}$$

$v_{\text{avg}}=\frac{L}{t_1}$

The average velocity in the first half is $\frac{L}{t_1}$.

(b) Determining her average velocities in the second half of the swim.

The average velocity of the person is in the second half is,

$$\begin{gathered} v_{avg}=\frac{\Delta x}{\Delta t} \\ =\frac{0-L}{t_2} \end{gathered}$$

$v_{avg}=-\frac{L}{t_2}$

Therefore, average velocity in the second half is $-\frac{L}{t_2}$.

(c) Determining her average velocities symbolically the round trip.

The average velocity of the person is in the round trip is,

$v_{avg}=\frac{\Delta x}{\Delta t}$

Since the person's total displacement on the round trip is zero, the person's average velocity will also be zero.

(d) Determining her average speed for the round trip.

The overall speed of a person is calculated by dividing the total distance travelled by the entire time taken. In a round trip, the person travels a total distance of$2L$.

The time taken by the person to cover this distance is$t_1+t_2$.

Thus, the speed of the person is,

$v_{avg}=\frac{2L}{t_1+t_2}$

Therefore, the required result is $\frac{2L}{t_1+t_2}$.