Question

Jack and Jill exercise in a 25.0 m long swimming pool. Jack swims 9 lengths of the pool in 155.5 s (2 min and 35.5 s) , whereas Jill, the faster swimmer, covers 10 lengths in the same time interval. Find the average velocity and average speed of each swimmer.

Answer 1

Answer:

Jack:

Average velocity: $\vec v_{avg, Ja} = +0.161\,\frac{m}{s}$, Average speed: $v_{avg,Ja} = 1.447\,\frac{m}{s}$

Jill:

Average velocity: $\vec v_{avg, Ji} = 0\,\frac{m}{s}$, Average speed: $v_{avg,Ja} = 1.608\,\frac{m}{s}$

Explanation:

We assume that each swimmer returns to the point of the departure each two lengths of the pool.

The average velocity is defined by the following definition:

$\vec v_{avg} = \frac{1}{\Delta t} \cdot \Sigma_{i=1}^{n} \vec r_{i}$ (1)

Where:

$\Delta t$ - Total time, measured in seconds.

$\vec r_{i}$ - i-th Relative vector position, measured in meters.

$n$ - Number of lengths done by the swimmer.

And the average speed is represented by the following formula:

$v_{avg} = \frac{1}{\Delta t} \cdot \|\vec r_{i}\|$ (2)

Where $\|\vec r_{i}\|$ is the norm of the i-th relative vector position, measured in meters.

Now, we proceed to calculate both values for both swimmers:

Jack - Average velocity ($\Delta t = 155.5\,s$, $\vec r_{1} = \vec r_{3} = \vec r_{5} = \vec r_{7} = \vec r_{9} = +25\,m$, $\vec r_{2} = \vec r_{4} = \vec r_{6} = \vec r_{8} = -25\,m$)

$\vec v_{avg,Ja} = \frac{5\cdot (25\,m)+4\cdot (-25\,m)}{155.5\,s}$

$\vec v_{avg, Ja} = +0.161\,\frac{m}{s}$

Jack - Average speed ($\Delta t = 155.5\,s$, $\|\vec r_{i}\| = 25\,m$, $n = 9$)

$v_{avg,Ja} = \frac{9\cdot (25\,m)}{155.5\,s}$

$v_{avg,Ja} = 1.447\,\frac{m}{s}$

Jill - Average velocity ($\Delta t = 155.5\,s$, $\vec r_{1} = \vec r_{3} = \vec r_{5} = \vec r_{7} = \vec r_{9} = +25\,m$, $\vec r_{2} = \vec r_{4} = \vec r_{6} = \vec r_{8} = \vec r_{10} = -25\,m$)

$\vec v_{avg,Ji} = \frac{5\cdot (25\,m)+5\cdot (-25\,m)}{155.5\,s}$

$\vec v_{avg, Ji} = 0\,\frac{m}{s}$

Jill - Average speed ($\Delta t = 155.5\,s$, $\|\vec r_{i}\| = 25\,m$, $n = 10$)

$v_{avg,Ji} = \frac{10\cdot (25\,m)}{155.5\,s}$

$v_{avg,Ja} = 1.608\,\frac{m}{s}$