Let s go through the mechanics of air resistance that way we can understand why an aero helmet may be a good option for someone.

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1 Shuld I wear an aer helmet? This is a great questin and ne n hand, the answer is easy: Yes, buy any equipment that will reduce air resistance during cycling. On the ther hand, the answer is mre cmplicated because mney may be better spent (r saved!) in ther ways. Let s g thrugh the mechanics f air resistance that way we can understand why an aer helmet may be a gd ptin fr smene. A gd way t understand air resistance is t cnsider terminal velcity f an bject drpped frm a high height and falling thrugh the air. Let s cnsider a situatin where gravity is the nly frce acting n the bject. When drpped, the bject has 0 m/s velcity and will start accelerating at a rate f -9.8 m/s/s (the acceleratin due t the frce f gravity). That means, after 1-secnd, the bject wuld be traveling m/s (remember velcity is a vectr and has magnitude and directin in this case the magnitude f velcity is 9.8 m/s and the directin is negative (r dwnward)). After 2-secnds, velcity wuld be m/s (i.e., velcity is increasing at a rate f -9.8 m/s every secnd while the bject is mving dwnward). The bject wuld cntinue t accelerate (i.e., speed up while mving dwnwards) as lng as gravity is the nly frce acting n the bject. In the real-wrld, there is anther frce acting n the bject air resistance. Air resistance is a frce applied t an bject based upn these factrs: - Velcity f the air mving ver the bject (v) - Density f the air (ρ) - Frntal area f the bject (A) - Cefficient f drag (C d ) In general, we talk abut air resistance as a frce that ppses the directin f mvement. Hwever, there are situatins where air resistance aids mvement fr example a tail wind during cycling (which is faster than the speed f the cyclist). Anther example wuld be sail bats r wind surfacing in these cases, the air resistance is prviding the prpulsive frce causing velcity. Hwever, fr ur discussin f air resistance during sprts like cycling and running, we will fcus n air resistance as a frce ppsing directin f mvement. Let s get back t terminal velcity When drpping an bject, that bject will increase velcity until the pint that the air resistance frce matches the frce due t gravity. At that pint, velcity becmes cnstant and the acceleratin f the bject is 0 m/s/s. Belw is a figure illustrating the tw frces acting n the bject as it is traveling dwnward: Gravity (pulling dwn) and air resistance (pushing up). We will start by using Newtn s 2 nd Law f Mtin.

2 F = ma F air F g = ma F g = F air at terminal velcity Therefre, a = 0 m/s/s F air = F g We can calculate the air resistance frce using this equatin: F air = ½ρv 2 C d A Let s put that in the equatin (remember F g = mg) F air = F g ½ρv 2 C d A = mg v = (2mg/ρC d A) 0.5 Where v is the terminal velcity f the bject. Using this equatin, that means that terminal velcity is a functin f Mass Acceleratin due t the frce f gravity (i.e., -9.8 m/s/s n Earth) Air density C d A cefficient f drag and frntal area That equatin tells us smething we already knw. Cnsider tw bjects, bth f the same size and shape but ne has mre mass than anther (belw) smething like a sccer ball vs. a bwling ball. The bwling ball will have a greater terminal velcity than the sccer ball.

3 Likewise, if we take tw bjects f different shapes, the ne that lks mre aerdynamic will have a greater terminal velcity than the nn-aerdynamic shape. The shape f the bject is represented in the equatin by the term C d A. Again, this is easy t understand that the air resistance f a big truck is mre than a small car simply due t the shape being mved thrugh the air. The pint f ging thrugh terminal velcity is that we can see hw easy we can link the velcity f the bject t the air resistance frce. The less air resistance frce, the greater the terminal velcity. Nw, let s g thrugh each f the parameters in the equatin t calculate air resistance. Velcity (v) Velcity represents hw fast the air is mving ver the bject. Cnsider running int a head wind, the velcity f the air mving past yu wuld be dependent n hw fast yu are running and hw fast the wind is blwing. T simplify the discussin, we ften just talk abut situatins where there is n wind then we can talk abut velcity relative t the grund. Velcity has units f m/s and when calculating the frce f air resistance, we take the square f velcity. That means that air resistance changes a lt based upn hw fast the bject is mving. If yu duble the velcity, yu increase air resistance by fur times! Density f air (ρ) Air has mass if we weigh a cubic meter (m 3 ) we wuld have abut kg f air (at sea level). When we talk abut density, we use units f mass per vlume, s air density wuld be kg/m 3. This can change based upn elevatin the higher the elevatin, the less dense air is. Temperature als influences air density the higher the temperature, the lwer the air density. Likewise, the amunt f water vapr present in air (i.e., humidity) influences air density the mre humid, the less dense air is. Cefficient f Drag (C d ) This parameter describes hw well air mves ver an bject. It is a functin f skin and frm drag where skin drag represents hw easy air mves ver a surface and frm drag is based upn the bject s shape. An bject that is mre aerdynamically shaped has a lwer frm drag.

4 Area (A) In this equatin, the Area (m 2 ) parameter represents the frntal area f the bject being mved thrugh the air. This is different than frm drag since frm drag is based upn the verall shape f the bject. In the case f Area, it is determined nly by the frntal area f the bject. In the picture belw, the Area wuld be getting smaller when lking at the pictures frm left t right. Many references will cmbine Cefficient f Drag and Area int ne term (C d A) because this then describes the verall influence f the shape being mved thrugh air. Ok nw let s talk abut cycling. Let s start with a kinetic analysis f cycling that is, let s identify all the frces (pushing and pulling) acting ON the bike/rider system. In the picture belw, there seems like a lt f arrws but let s reduce them t these: - Frces ppsing mtin: Air resistance (Fair) Bearing resistance (Fb) Rlling resistance (Fr) Slpe (this will be explained mre belw) (Fs) - Frce causing mtin: Prpulsin (Fp)

5 Let s g thrugh each f these parameters. Air resistance (Fair) This was cvered abve in the discussin f terminal velcity same frce. The imprtant part here is t recgnize that the term C d A is influenced by psitin f the cyclist as well as the helmet, clthing, frame shape, and placement f water bttles. A mre upright rider will have greater air resistance than a rider in an aer psitin, fr example. Bearing resistance (Fb) This is the resistance f the bearings that allw the wheels t spin. Cnsider just hlding the bike ff the grund and giving a wheel a spin eventually the wheel will stp rtating. The resistance that causes the rtatin is stp is referred t as Bearing Resistance. This is nt that big f a frce as cmpared t sme f the ther frces. Nevertheless, cyclists will ften pay clse attentin t the hubs that a bike has in rder t reduce the bearing resistance. Rlling resistance (Fr) This is the resistance that cmes frm the tire rlling alng the grund. It s nt hard t imagine that biking in sand is harder than biking n a paved rad. The difference in this is because f rlling resistance. As the tire rtates and cmes int cntact with the grund, the tire defrms. As the wheel cntinues t rtate, the tire rebunds but there is always energy lst in this prcess. Rlling resistance is calculated using this frmula: Fr = C rr N where C rr is called the Cefficient f rlling resistance and N is the nrmal (i.e., vertical) frce. C rr is a dimensinless parameter and describes hw well a tire rlls ver the grund. A tire with a lw C rr des nt rll very easily. When a cyclist purchases tires, he/she will ften ask what is the C rr? Yu can als see frm the equatin that the Fr is influenced by bike/rider weight (i.e., N). This is ne reasn why cyclists try t keep their bike and rider weight as lw as pssible. The test f Crr is dne n a smth surface. Hwever, Fr can be increased if we are riding n a bumpy surface. The tire may r may nt defrm ver the bump. If it des, energy can be lst in the defrmatinrecil prcess. If it des nt, the wheel is lifted vertically which influences the resistance f rlling frward. Sme references develp a separate term fr bumps whereas thers include it in Fr. Slpe We all knw cycling uphill is harder than cycling n a flat rad. The reasn fr that is that when we ride up a slpe, a cmpnent f gravity ends up ppsing the directin f mtin (r, if yu are riding dwn hill, the cmpnent f gravity aids mtin). We calculate slpe frce using this equatin: Fs = (mg)sin( )

6 where mg is the weight f the rider/bike system and is the angle f the slpe. This is anther reasn cyclist try t keep the rider/bike weight lw the bigger this weight (i.e., mg), the harder it is t ride up hills. Prpulsive frce (Fp) This is the frce that makes the bike g this is a result f the cyclist pushing n the pedals. That turns the cranks, which causes the chain t pull n the rear wheel, which then pushes n the grund t cause prpulsin. Even thugh there are a lt f places that energy can be lst between what the rider des and the pint where the wheel causes prpulsin, we can cnsider that Fp is caused by the rider (the efficiency f bike transmissins may be abut 80-95% which is pretty gd!). Thinking like a Bimechanist Nw let s pull tgether the different pieces and try t see the basis fr an answer t the questin: Shuld I buy an aer helmet? Once again, we start with Newtn s 2 nd Law f Mtin: F = ma Fp + (-Fair) + (-Fr) + + (-Fb) + (-Fs) = ma Prpulsin frm wheel (Fp) Air resistance (Fair) Rlling resistance (Fr) Bearing resistance (Fb) Slpe resistance (Fs) Cnsider riding at cnstant velcity (which is what we are trying t d during an endurance event), then a = 0 m/s/s therefre: Fp = Fair + Fr + Fb + Fs Nw we see that the prpulsin frce is a functin f the resistance frces. That is, the rider needs t create mre prpulsin frce in rder t vercme a greater air resistance, riding up hill, r greater rlling and bearing resistance. Shuld a cyclist buy an aer helmet? The easy answer is yes the cyclist shuld buy anything t reduce the resistive frces. The mre cmplicated answer is that if a rider purchased every piece f available equipment t reduce thse frces, the bike wuld likely cst ver $15,000. Riders need t make infrmed chices and match the equipment with his/her capabilities.