Throwing, Jumping and SpinningThrowing, Jumping and Spinning

John D BarrowJohn D Barrow

Launch upwards with a vertical speed V from a height H0 then H = HH = H00 +Vt – ½ gt +Vt – ½ gt22 after a time t, after a time t,

g = 9.8 m/s2 is acceleration due to gravity. It returns to the thrower’s hand at the same height, H = HH = H00,

after a time tthh = 2V/g = 2V/g. If it was tossed upwards with a spin of R revolutions

per sec then it will have turned over completely N times where

N = tN = thh × R = 2VR/g × R = 2VR/g 0.4s 0.4s R R

ififV V 2 m/s, g = 9.8 m/s 2 m/s, g = 9.8 m/s22

Coin Tossing Isn’t RandomCoin Tossing Isn’t Random

Need R > 50/sNeed R > 50/sfor N > 20for N > 20

N = 1 and the coin was launched very slowly, with N = 1 and the coin was launched very slowly, with heads upwards then it will fall with heads up againheads upwards then it will fall with heads up again

N = 2 to 3, or 4 to 5, or 6 to 7, etc, it will be caught with N = 2 to 3, or 4 to 5, or 6 to 7, etc, it will be caught with the same face up as it was thrown with.the same face up as it was thrown with.

N = 3 to 4, or 5 to 6, or 7 to 8, etc, it will land with the N = 3 to 4, or 5 to 6, or 7 to 8, etc, it will land with the opposite face upwardsopposite face upwards

N >> 20 is large: the conditions on V and R that N >> 20 is large: the conditions on V and R that distinguish the two outcomes get closer and closer distinguish the two outcomes get closer and closer together and very small differences in the tossing together and very small differences in the tossing conditions result in heads or tailsconditions result in heads or tails

In general, if n = 1,2,3,4,….etc then if N lies between 2n In general, if n = 1,2,3,4,….etc then if N lies between 2n and 2n+1 the coin will land with the initial face upwards and 2n+1 the coin will land with the initial face upwards and if n lies between 2n+1 and 2n+2 the initial face will and if n lies between 2n+1 and 2n+2 the initial face will land downwardsland downwards

More Revolutions Means More UnpredictabilityMore Revolutions Means More Unpredictability

Biased Coins Don’t Biased Coins Don’t HelpHelp

They always spin about an axis that passes They always spin about an axis that passes through the centre of mass even when through the centre of mass even when centre of mass centre of mass geometrical centre geometrical centre

Skaters can reduce their inertia (Skaters can reduce their inertia ( Mr Mr22) ) by a factor of two and this will double the by a factor of two and this will double the

angular speed of a spin up to about angular speed of a spin up to about = 20 radians/ s, or 3 revs/ s.= 20 radians/ s, or 3 revs/ s.

Skating SpinsSkating Spins

MrMr22 = constant = constant

1/r1/r22

r

Wheeler DealingWheeler Dealing

Total energy needed to move the bike frame + two wheels Total energy needed to move the bike frame + two wheels

at speed v and rotate the two wheels at at speed v and rotate the two wheels at = v/r = v/r is is

Total energy = ½ (2m + mTotal energy = ½ (2m + mframeframe)v)v

22 + 2 + 2 ½ I ½ I22

Since I = bmrSince I = bmr22 with b with b 1 (ring) 1 (ring) ½ (disc) ½ (disc)Total energy = ½vTotal energy = ½v22{m{m

frameframe + 2(1+b)m} + 2(1+b)m}

Wheels On FireWheels On Fire

The same mass reduction on each wheel is three or four timesThe same mass reduction on each wheel is three or four times as beneficial as the same mass reduction on the frame.as beneficial as the same mass reduction on the frame.

Momentof

inertia

The Grand JetéThe Grand Jeté

The centre of mass always follows a parabolic trajectory after take-offThe centre of mass always follows a parabolic trajectory after take-offThere is nothing the dancer can do to alter that.There is nothing the dancer can do to alter that.

So is the appearance of floating a fiction?So is the appearance of floating a fiction?

You can change your body shape so that the trajectory of your headYou can change your body shape so that the trajectory of your headdoes not follow the same trajectory as your centre of massdoes not follow the same trajectory as your centre of mass

Throw a mallet or a shuttlecock or a caberThrow a mallet or a shuttlecock or a caber

Change Your Mass DistributionChange Your Mass Distribution

In the jump phase, raise legs to horizontal + put arms above shoulders. Raises CoM relative to head.

Then lower legs and arms and CoM falls in descentRelative to head, her CoM is shifting upwards

It follows parabola but the head stays horizontal

Height of the dancer’s head above the floorHeight of the dancer’s head above the floorDoes not follow a parabolic pathDoes not follow a parabolic path

Basketball players’ illusion of ‘hanging in the airBasketball players’ illusion of ‘hanging in the air ’’

The Cat ParadoxThe Cat Paradox

1. Bend in the middle: the front and back halvesrotate about a different axes.

2. Tuck in front legs: reduce moment of inertia of the front of body and extend rear legs to increase the inertia of the rear half, so can rotate front half as much as 90° while rear half rotates in the opposite direction much less -- as little as 10°.

3. Extend front legs and tuck in rear legs: rotate rear half a lot while the front half rotates much less in the opposite direction.

4. Depending on its flexibility and initial angular momentum, the cat may need to repeat steps two and three times to complete a full 180° rotation.

‘‘II = constant’ = constant’

Aim: Enter the water vertical with no rotationAim: Enter the water vertical with no rotation Increase your inertia so as to spin slowerIncrease your inertia so as to spin slower

Another Irrotational LanderAnother Irrotational Lander

Diving – how much time have you got ?Diving – how much time have you got ?High board h = 10mHigh board h = 10m

h = ½ gth = ½ gt22 t = t = (20/9.8) s = 1.4 s(20/9.8) s = 1.4 sV = gt V = gt 14 m/s = 31 mph 14 m/s = 31 mph

Three and a half somersaults in the air means 3.5/1.4 = 2.5 rev/s Three and a half somersaults in the air means 3.5/1.4 = 2.5 rev/s This is 150 revolutions per minute (rpm) This is 150 revolutions per minute (rpm)

The tracking speed at the outer edge of a disc on your CD player is 200 rpm. The tracking speed at the outer edge of a disc on your CD player is 200 rpm. Each revolution is 2Each revolution is 2 radians so the angular velocity is 2 radians so the angular velocity is 2 2.5 = 5 2.5 = 5 = 15.7 radians/s = 15.7 radians/s

Springboard Divers Have More TimeSpringboard Divers Have More Time

t t 1.8s in the air 1.8s in the air5 deg to vertical5 deg to vertical

This is vitalThis is vitalOr…Or…

SpringboardsSpringboards

Hit the board at its ‘natural’ frequency to create a resonanceHit the board at its ‘natural’ frequency to create a resonanceYou can hear itYou can hear it when it’s right (or wrong)when it’s right (or wrong)

xx

Restoring force = - kx = m dRestoring force = - kx = m d22x/dtx/dt22

x = 0.75mx = 0.75m deflection standing still means kx = mg deflection standing still means kx = mg k = 65k = 659.8/0.75 = 848 N/ m9.8/0.75 = 848 N/ m‘‘Natural’ frequency= Natural’ frequency= (k/m) ~ (k/m) ~ 848/65 = 3.6/s for m = 65 kg diver848/65 = 3.6/s for m = 65 kg diver

More board out to the right of roller More board out to the right of roller ‘springier’ ‘springier’ large k large k

4.9m x 50cm4.9m x 50cm aluminiumaluminiumcantilevercantilever

Eg: Length of jump = A + B + C Eg: Length of jump = A + B + C 0.4m + 6.45m + 0.9m = 7.75m 0.4m + 6.45m + 0.9m = 7.75mFor V For V 9 m/s and 9 m/s and 20 deg 20 deg

B = (VB = (V22/g) cos/g) cos [sin [sin + (sin + (sin22 + 2gH/V + 2gH/V22))1/21/2]]

Anatomy of A Long JumpAnatomy of A Long Jump

It is up to the jumper where the board is on take-offIt is up to the jumper where the board is on take-off

hanghang

sailsail

Hitch kickHitch kick

Styles that Aid Forward RotationStyles that Aid Forward Rotation

Bent leg at take-off for lowerM of inertia, faster free leg rotnDrive opposite arm upwards

Move limbs for better landing posn.Counter forward rotn from

Leg thrust behind CoM at take-off

Delay body’s forward speedAllow time for legs to positionControl rotation by long thin shape in the air 1.5 or 2.5 ‘strides’ in the air

Doesn’t counter forward rotn at take-off

Sideways fallouton landing low rotn

Ang mom of bent andstraight legs balance

Kicking for Time Rather Than DistanceKicking for Time Rather Than Distance

R = VR = V22/g /g sin(2 sin(2))sinA =sin(180 – A)sinA =sin(180 – A)

So R is same for launch angles So R is same for launch angles and 90 - and 90 - Time in the air =Time in the air = R/(VcosR/(Vcos))

t(high)/t(low) = cos(t(high)/t(low) = cos()/cos(90-)/cos(90-) = cot() = cot() ) Height achieved is H(high)/h(low) = cotHeight achieved is H(high)/h(low) = cot22(().).

eg t(eg t( = 75 deg = 75 deg)/t()/t( = 15 deg = 15 deg) = 3.7) = 3.7H(H( = 75 deg = 75 deg)/H()/H( = 15 deg = 15 deg) = 14) = 14

Javelin is launched in a flatter trajectory of around 35 degrees It hangs in the air for longer, and sails out to a greater distance. javelin is angled to lie at about 10 degrees (the ‘angle of attack’) below the trajectory followed by its centre of massThis ensures that the nose of the javelin is always dropping

Javelin ThrowingJavelin Throwing

Centre of pressureCentre of pressure

Centre of massCentre of mass

An overturning momentAn overturning moment

changes angle of attackchanges angle of attack

C o P is first behind, thenC o P is first behind, then ahead, then behind C o Mahead, then behind C o M

dragdrag

liftlift

When the lift exceeds drag it will be larger when throwingWhen the lift exceeds drag it will be larger when throwinginto a wind because it is proportional to (V– W)into a wind because it is proportional to (V– W)22 and so it will be much and so it will be much larger with a headwind (Wlarger with a headwind (W

< 0 ) than with a following wind (W > 0). < 0 ) than with a following wind (W > 0).

A good situation is to throw into a 10 m/s headwind, where you will A good situation is to throw into a 10 m/s headwind, where you will throw about 4m further than in still air. This is a very strong wind!throw about 4m further than in still air. This is a very strong wind! With a 10 m/s tailwind your throw is reduced by about 2m. With a 10 m/s tailwind your throw is reduced by about 2m. A tailwind of 7.5 m/s is the worst case scenario. A tailwind of 7.5 m/s is the worst case scenario.

FFdragdrag C(V-W) C(V-W)22

FFliftlift D(V-W) D(V-W)22

The Discus -- Lift can Beat DragThe Discus -- Lift can Beat DragDistance thrown

A headwind can help the discus throwerA headwind can help the discus thrower

Bend It Like A BeckhamBend It Like A Beckham

No Spin On the BallNo Spin On the Ball

Speed Speed as Pressure as Pressure

Net air speed near the ball is less at top than bottomNet air speed near the ball is less at top than bottomPressure on the ball at the top is bigger than at the bottomPressure on the ball at the top is bigger than at the bottom

This creates a downward forceThis creates a downward force

Spinning BallSpinning Ball

An In-swinging CornerAn In-swinging Corner

The Archer’s ParadoxThe Archer’s Paradox

The Oscillations of the ArrowThe Oscillations of the Arrow

Time sinceTime sinceArrow’sArrow’sreleaserelease

No part of the arrowNo part of the arrowshould touch the bowshould touch the bowas it passes byas it passes by

The Stiffness (Spine) of the Arrow is CrucialThe Stiffness (Spine) of the Arrow is Crucial