The Physics of Springboard and Platform Diving
Physics 24
UNC-CH

Diving terminology:
- the approach- An approach is the diver's walk down the board where thery press the board on the last step.  The "press" is an exaggerated step which, along
with the walk, create an oscillation and rhythm in the board.  An approach typically has four steps.
- the hurdle - The hurdle occurs after the last step.  It is when the diver leaps from one leg onto two legs, landing on the end of the board.

Types of dives:
- forward - forward takeoff with forward rotation
- backward - backward takeoff with forward rotation
- reverse - forward takeoff with backward rotation
- inward - backward takeoff with forward rotation
- twister - one of the four above groups with twisting in 1/2 twist increments (1/2 to 4 twists)

Position of dives:
- straight- This position is when a diver's body is as fully extended and rigid as possible.  Because it has the greatest possible radius, the fewest somersaults are
possible.

- pike - This position is when the body is bent only at the hips, with legs straight and arms and head by their ankles.  This position has a smaller radius than the
straight position, making somersaulting easier.

- tuck - This position is then the body is bent at the hips and the knees, creating the smallest radius possible.  This position has the largest possible angular
acceleration and therefore the greatest number of somersaults are possible in the tuck position.

- free - This position is a combination of the three above positions, used only in twisting dives where multiple positions are required at different parts of the dive.
While somersaulting, the dive may be tuck or pike; while twisting, the body must be straight.

Judging:  Diving competitions can have five to nine judges.  Dives are rated from 0 to 10, 10 being perfect and 0 being incorrect dive.  A dive may considered
incorrect or incomplete if you do the wrong dive (front 1.5 tuck instead of back 1.5 tuck) or if the wrong part of the diver's body hits the water first.  In
a dive landing on the head, the hands must enter the water first.  In a dive landing on the feet, the feet must enter first.  Dives are scored with digital or
manual score cards in increments of 0.5.

Each dive has its own degree of difficulty based on the position, which board height it is preformed from, and the number of rotations and twists.  For each dive, the highest and lowest scores of the judges are discarded.  The sum of the other scores are added and multiplied by the degree of difficulty, which is larger for more difficult dives.  Therefore, divers are rewarded for learning more difficult dives, but not enough that the finesse of the dive can be discounted.

The diving board:
In international competition, a Maxiflex diving board is used.  Created and produced by Duraflex, the board has an aluminum I-beam center.  It is 16 feet long, 19 5/8 inches wide and 1 5/8 inches thick.    The website for more information on Duraflex is available at www.springboardsandmore.com.

The Physics of Springboard and Platform Diving

Rotation and Inertia

Before applying physics concepts to diving, a few basic terms must be understood.

moment of inertia ( I ) - It is the measure of the distribution of the object's mass about the axis of rotation.  I = mr2.

- angular velocity ( w )- It is the speed of rotation about the axis of rotation.  w = change in angle / change in time

- angular momentum - It is the quantity of rotation a body has about a given axis as a result of its speed of rotation and the distribution of mass about its
axis.    Angular momentum = Iw

- angular acceleration ( a ) - It is the change in angular velocity over time.  a = w / change in t.

- torque ( T ) - It is the tendency of a force to rotate an object about some axis.   T = Fd, where F is the force applied and d is the distance.  Also, T = Ia.

QuestionDoes a diver break the law of conservation of angular momentum?

Answer: Once the diver leaves the diving board, his angular speed may increase or decrease.  This does not violate conservation of angular momentum because the moment of inertia will vary also.  The equation for angular momentum conservation is  Iiwi = Ifwf .  Becuase no torque is applied to the diver once he leaves the diving board, momentum is conserved.  When the angular speed increases due to changing his body's radius, the moment of inertia subsequently decreases, thus keeping the equation constant.

Question: In many dives, divers begin twisting after their feet have left the board.  With no object to push off from, how is this possible if the net torque equals 0?

Answer:By changing the radius, the moment of inertia must change due to the equation I = mr2.  Because we know that the net torque, moment of inertia, and angular acceleration are related by T = Ia, it is seen that the angular acceleration must change if no net torque exists.  To increase the speed of the somersaults, a diver curls up into a tight tuck position, thus drastically reducing his radius.  For T = 0, the angular acceleration must increase.  After completing the desired somersaults, the diver extends his arms and legs, thus increasing his radius and decreasing the angular acceleration.  I increases as the square of the distance of the mass.  Therefore, a small increase in the distance can result in a relatively large increase in I.

Spring Constant and its Effect on Springboard Diving

Question: How much higher can you jump on a springboard as opposed to the ground?

Answer:If the person weighs 150lbs, their crouch height (the amount they bend down before attempting to jump) is approximately 0.4m; the time it takes for their body to extend is approximately 0.25s (this is the speed at which muscle can contract).  They are then able to jump about 0.52m (1.7ft) off the ground.  If a 150lb (68kg) person is jumping off a springboard with a spring constant (k) of 833N/m, then they are able to jump to a height of 0.92m (3.0ft) off the board.  So, using the springboard can almost double the liftoff height of a diver!!!  In addition, a springboard is constructed with a fulcrum (as seen in picture below).  Located in approximately the middle of the board (closer to the stand then to the tip of the board), the spring constant is fairly large (more resistance to force), thereby decreasing the possible distance of board depression.  However, if you move the fulcrum closer to the stand, the distance the board can be depressed will increase and the spring constant will decrease (less resistance to force).  Therefore, the height the diver can jump from the board will also increase. In fact, if the distance the board can be depressed by a 150lb person increases to 1.2m (by moving the fulcrum closer to the stand) the height will increase from 0.92m to 1.2m.  This is almost a 1 foot  increase.  Also, a diver uses an approach before they take off for a dive.  In the hurdle, they initially jump up then land on the tip of the board.  This creates a higher force acting on the board, which will in turn increase the height of the jump.

Calculation for jumping on the ground:
Principle of Conservation of Energy: PEi+KEi=PEf+KEf
PEi = 0 and KEf = 0 so,  KEi = PEf
KEi = 1/2 mvcm and PEf = mgH
1/2 mvcm= mgH
H = vcm2 /2g
vcm = velocity of the center of mass of the jumper = 2h/t  h=crouch height and t=extension time
vcm = 2(0.40m)/0.25s = 3.2m/s
H = (3.2m/s)2/(2)(9.8m/s2)
H = 0.52m (for a 150lb male)

Calculation for jumping on a springboard:
Principle of Conservation of Energy: PEi+KEi = PEf+KEf
KEf = 0 but PEi = 1/2 kx2 because there is stored elastic potential energy in the spring, whereas when you jump off the
ground there is no initial gravitational potential energy because the earth does not provide any
significant source of potential energy to the jumper.
So, PEi + KEi = PEf
1/2kx2 + 1/2mvcm2 = mgH
H = (1/2kx2+1/2mvcm2)/mg
F = kx      k=spring constant and x=distance springboard is depressed with a 150lb person on it
k = F/x
k = ((68kg)(9.8m/s2))/0.8m
k = 833N/m
H = ((1/2(833N/m)(0.8m)2) + (1/2(68kg)(3.2m/s)2))/(68kg)(9.8m/s2)
H = 0.92m (for a 150lb male)

The depression of the diving board during a dive is pictured below.  As can be seen from the photo, the board can be depressed quite a
bit by the weight of the diver, especially when compared to the empty board next to it.  More photos and information about the duraflex
diving boards can be found on the Duraflex International website.

“Play-time Pool Antics”  ~A closer look at projectile motion and velocity

Taking a less serious approach at the diving scene, focusing more on the possibilities of playing around is what most kids do. Kids like to create their own competitions among each other, seeing who is the best.  One of the things kids do at the pool is see who can jump the farthest. Applying physics to this can enable one to see what conditions actually affect this distance. This is a projectile motion problem, making two equations necessary.

Dx = vxot                   and                   Dy = vyot – 1/2gt2
The first equation is the horizontal motion component.  The x component of velocity is the same as the kid’s velocity right before jumping, vxo. Because vxo is zero, we find x = vxot and solve for t.  Once we know the time it takes to hit the water, we can find Dx along the horizontal.
For t, the equation for vertical motion is used.  If we know how high the concrete surface is above the water, and we know initial velocity in the y direction, we need only to plug into an equation.  The ledge will be assumed to be approximately one foot off of the surface of the water, approximately 0.3m.   Because we know vi in the y direction to be zero before leaving the edge, the kids velocity is only in the horizontal direction thus far. Therefore
y = -1/2gt2
-0.3m = -1/2(9.80m/s2)t2
t in air = .0247s

Because kids usually don’t have much room around a pool to gain lots of speed, a velocity estimate of 3 m/s will suffice for the next part. With this estimate and t, we can now plug into the 1st equation to find Dx.
x = vxot
x = (3m/s)(0.247s)
x = 0.74m

This is not very far! This approximation isn’t the most accurate, however, because one usually jumps up and then out, which extends Dx. In this case, an equation such as Dx = vxot = (vocosQo)t will be used to better approximate.  We can, however, explore the contributions that each parameter makes to this horizontal distance jumped. For instance, if one were to jump from a height of 5 meters, this would then affect time considerably, resulting in a 3m horizontal jump.  Similarly, increasing another part, such as velocity, gets different results.  Jumping from the pool’s ledge at 9m/s, as opposed to 3m/s, x will equal a horizontal 2.2m.  Try jumping at 9m/s from 5m, and the horizontal distance shoots to 9m!

Another popular thing to do is see how high one can climb before jumping.  Kids get a rush out of jumping from high structures.  Lots of recreation pools have different platforms, or a high dive platform, some similar to those found in Olympic dive competition settings.  So how much does height actually affect velocity?
To solve this, two equations are again necessary.
y = -1/2gt2 and vy = vyo – gt
If one jumps from 15 meters, plugging this into the 1st equation, t can be found.
-15m = -1/2(9.8m/s2)t2
t = 1.75s

Now plugging t into the 2nd equation
vy = 0 – gt
vy = -(9.8m/s2)(1.75s)
vy = 17m/s

For the not-so-brave kid, 15m is pretty high! Scaling down to 10m, using the same equations and process to solve, the velocity reduces from 17m/s to 14m/s.  And if you were one of those kids who couldn’t handle height at all, there was always that platform just right above the water around 1m. Again, using the same solution process, one finds the final velocity before entering the water to be 4.41m/s – not too scary.
So it turns out that the higher you climb, the faster you’ll drop! This method of approximation is also not quite accurate if given varying conditions. Most pools have springboards, which would increase Dy.  Jumping up, increasing this Dy, contributes to a slightly larger vy.

Divers must rotate pretty quickly to do intricate, multiple flips and sommersaults before entering the water. This is another enticing game for children. Just how many flips can one do before they hit the water?  For starters, one must consider not only how much time one has (which we will discuss here), but also how long the average flip takes.  This is one reason belly flops and awkward landings happen so much.  One can only fit so many flips or tricks in safely before landing.
So just how much time does one have before landing? To answer this question, one needs only know how high he is, and one equation.
Dy = vyot – 1/2gt2
If jumping from a 10m platform
-10m = -1/2(9.8m/s2)t2
t = 1.43s
If jumping from a 3m springboard or platform (eliminating the extra height from the springboard for simplicity’s sake), this time becomes 0.78s.

Just looking at these questions, it becomes clear that adjusting only one thing in the equation can make all the difference in the result. Physics works ~ playtime is governed by rules after all!