      # Rowing power and kinetic energy Here we continue the analysis of rowing power and its conversion into kinetic energy of the rower-boat system. The definitions and measurements data from RBN 2019/01-02 is used here. The main equations of the rower-boat system movement relate forces applied to its components with their masses and accelerations. The total force applied to the boat Fb is a difference between its propulsive Fbp and drag Fdr forces:

Fb = Fbp - Fdr =Fpx â€“ Fsx - Fdr = mb ab               (1)

where Fpx horizontal (X) component of the pin force, Fsx - X stretcher force, mb ab â€“ product of boat mass and acceleration. Contrarily to the boat, which is a passive component in this case, the rower is an active part of the system: they applies forces to rowing equipment. So, rowerâ€™s mass is driven by reaction forces, which have the same magnitude and opposite direction to action forces: a rower pushes the stretcher backwards and reaction force Fâ€™sx accelerates their CM forward, pulling the handle and the reaction force Fâ€™hx moves them backwards. The resultant force applied to the rowerâ€™s CM Fr is also a difference between their propulsive force Frp and drag (air resistance) Frr, which is quite small and can be neglected:

Fr= Frp â€“ Frr ~ Fâ€™sxâ€“ Fâ€™hx = mr ar                                    (2)

where mr ar are rowerâ€™s mass and acceleration. The force applied to the whole system is the sum of rowerâ€™s and boat forces, so using Eq.1 and 2:

Fs = Fb + Fr = (Fpx â€“ Fsx - Fdr) + (Fâ€™sxâ€“ Fâ€™hx) =

= Fpx â€“ Fâ€™hx - Fdr  = ms as                                               (3)

Having forces F applied to the boat, rowers and the whole system CM (Fig.1,a) and their masses m, we can derive their accelerations a (Fig.1,b, the total forces must be used here including the drag):

a = F / m                                                              (4)

Velocities v can be obtained as an integral of accelerations plus a constant offset voff to make an average velocity over the stroke cycle equal to zero:

vLi =vLi-1 + a dt + voff                                            (5)

The velocities vL (Fig.1,c) are represented in the local inertial reference frame, which moves with constant velocity equal to the average rowing velocity vav over the stroke cycle. In mechanics, this frame is called the â€ścentre of momentum frameâ€ť: â€śThe kinetic energy of systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the centre of the momentum frame, i.e. the reference frame in which the total momentum of the system is zero. This minimum kinetic energy contributes to the invariant mass of the system as a whole.â€ť

To derive absolute velocities vG in global reference frame based on the water, the average rowing velocity over the cycle vav should be added to the local velocities vL:

As we discussed before (see RBN 2018/11), there are two possible scenarios here:

Â·       Internal power transfer between components of the system, where velocities in the local system frame must be used;

Â·       External power exchange with environment, where velocities in global reference frame must be used.

Internal (local) power can be derived as products of their propulsive forces (excluding boat drag) and local velocities (PL=FpvL, Fig.1,d), and internal kinetic energy can be found with masses of the components (EkL=0.5mvL2, Fig.1,e). We will discuss internal energy first. During the recovery, the oar is disconnected from the water, so there is no propulsive force, and the only external force is the drag applied to the boat hull, and so to the system. All other forces are internal ones, so their power should be derived with velocity in the local reference frame. Neglecting the small rowerâ€™s drag force, power applied to their CM in this local frame PrL is:

PrL = Fr vrL = (Fâ€™sxâ€“ Fâ€™hx) vrL                     (7)

The local boat power PbL is:

PbL = Fbp vbL = (Fpx â€“ Fsx) vbL                            (8)

The power of the system PsL can be derived in two ways: 1) The sum of the rowerâ€™s and boat powers PsInt shows energy spent on internal movements of the components within the system:

PsInt = PrL + PbL                                                   (9)

2) Product of the force and velocity at the system CM defines movement of the system as a whole piece, but it valid only in the global frame, as only external forces applied to it:

PsG = Fsp vsG = (Fpx â€“ Fâ€™hx) vsG               (10)

During recovery, both rowerâ€™s and boat powers are positive (1), because their forces and local velocities are in the same direction: for the rower, they are negative (they pull themselves backwards through the stretcher, and their CM also moves backwards in the local frame); boat force and local velocity are positive (it is pulled forward through the stretcher, and its velocity is higher than average).

Generally, positive power means concentric muscle contraction: force and velocity work in the same direction (both positive or both negative, depending on the reference frame) and muscles produce mechanical energy. Negative power means eccentric muscle contraction: force and velocity act in opposite direction, so the muscles consume mechanical energy produced elsewhere.

Before the catch, the boat power gets close to zero for a short time (2), because both its force and local velocity change direction. Then, the boat power increases again (3), because both the force and local velocity obtain negative direction at the same time. The rowerâ€™s internal power becomes negative before the catch (4), because the force becomes positive, but local velocity is still negative and only starts increasing slightly: the rowerâ€™s mass is much heavier and canâ€™t change direction as quickly as the boat. Here, we have a typical example of collision of two masses.

From mechanics, there are two extreme kinds of collisions: elastic ones, like billiard balls, which bounce in opposite directions after collision with nearly the same speed as before, and inelastic - like two pieces of clay, which stick to each other and moves together as one piece after collision. In both cases, the total momentum of the system is not changed as it is dictated by â€śMomentum conservationâ€ť law. However, internal kinetic energy of the system (sum of kinetic energies of the components) becomes very different:

Â·                 In a perfectly elastic collision, the objects exchange their kinetic energies, so internal energy of the system is preserved (energy is a scalar quantity, not a vector).

Â·                 In inelastic collision, the kinetic energy is consumed for deformation of the objects and dissipated as heat, so the internal energy of the system becomes zero: there is no internal movement in the system after the collision.

Of course, there are no perfectly elastic collisions in pure mechanics, as part of the energy is always consumed in deformations, but in the real world, energy can be added with muscular contraction. Biomechanics studies in top sprint runners revealed that up to 90% of take-off energy is delivered by the elastic properties of tendons, and only 10% is added by muscles. Of course, movements are much slower and more complex in rowing, so the above efficiency looks unrealistic. However, the more elastic collision happens between rowerâ€™s and boat masses at catch, the more internal kinetic energy is preserved, so it can be recycled during the drive phase and rowing efficiency increases. This means an effective catch should look like â€śa bouncing ballâ€ť. This perfectly confirms our previous concepts of effective rowing: â€ścatch through the stretcherâ€ť, â€śtrampoline effectâ€ť and â€śhammer and nail principleâ€ť, and adds one more argument against older concepts of â€śminimising boat checkâ€ť, â€śdo not upset boat at catchâ€ť, etc. Fig.1,e shows that the boat kinetic energy after the catch (5) becomes higher than during the recovery, which means technique of this sculler is quite efficient.

If the rowerâ€™s and boat masses collide inelastically at catch, a rower has to consume kinetic energy during the recovery, which fatigues their muscles anyway, then produces it again with muscles to start the drive, so efficiency suffers. The worst thing a rower can do is to slow down seat movement during the second half of recovery and â€śstick to the stretcherâ€ť at the catch.

During the drive, the picture is much more complicated, because external power from the oar is added and mixed up with internal power, so we put question marks on the drive phase of Fig.1,d. External powers of the system components can be derived with equations similar to Eq.7-10, where global velocities are used instead of local ones (Fig.2,a). However, during the recovery, rowerâ€™s and boat global propulsive powers are not valid as they are moved by internal stretcher force mainly, so they are crossed out. The global system power can be derived in two ways: PsG(r+b) is the sum of the rowerâ€™s and boat powers; and PsG with the propulsive force FsP and velocity vsG:

PsG = FsP vsG = (Fpx â€“ Fâ€™hx ) vsG             (11) Fig.2,b compares both system powers above with traditional handle power Phnd and sum of the body segments power Psegm (RBN 2004/06). Average power PsG(r+b) was quite similar to Phnd (Â±3% at 17-41spm), PsG was similar to Psegm (Â±5%) and the range of the system kinetic energy variation was close to the measured work per stroke (Â±2%).

Fig.2,c compares the internal system power from Fig.1.d (on the left scale) with differences between global kinetic power and rowerâ€™s power production (on the right scale). The curves look very similar, which suggests a hypothesis explaining the second ones with an effect of the first ones, but the problem is that the magnitudes of global differences were 4-5 times higher than local powers.

We will continue the analysis in the next bulleting, all comments and ideas are welcome.