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**1. Swimming efficiency,** or also called stroke index, is an important concept for improving performance in the water. There are several formulas for calculating swimming efficiency, but let's look at two common approaches:

Swimming Equation: The swimming equation is a formula that considers variables such as reaction time, time under water, turn time, stroke count, and stroke frequency

Here's how it's written: [ ST = S + (UT + TT) + (CC \cdot SR) ]

- (ST): Swimming time, in seconds..
- (S): Start (Reaction Time + Time in Air), in seconds.
- (UT):Time underwater, i.e. how much time you spend underwater after starting or turning.
- (TT): Time underwater, i.e. how much time you spend underwater after starting or turning.
- (CC): Number of strokes.
- (SR): Stroke rate, in seconds per stroke.

This equation represents two components: time under water and time above water. Time under water is the sum of (UT) and (TT), while time above water is a function of (CC) multiplied by (SR)..

2. Speed Formula: Another approach involves swimming speed.

**The formula is:** [ v = \frac{{P_m \cdot E_p}}{D} ]

- (v): Swimming speed.
- (P_m): Muscle power.
- (E_p): Propulsive efficiency.
- (D): Water resistance.
- cdot: represents the multiplication operation between the various terms of the formula ( cdot indicates that the metabolic power developed by the swimmer (( P_m )) is multiplied by the propulsive efficiency of the swimmer (( E_p )), and the product of these two terms is then divided by the drag (( D )), which represents the resistance of the water.

This formula takes into account muscle power, propulsive efficiency and water resistance to calculate swimming speed.

In short, improving swimming efficiency requires good technique, correct stroke frequency, and effective time management under and above the water.