Someone in my WhatsApp group sent me a video
of bungee jumping for beginners accompanied by a lady trainer who hugs the
jumper during the dive.
After
seeing the video, I asked myself in a bungee jump from a height, don’t the
jumper suffer from injuries similar to him falling on the ground. Let us try to
examine this question to get the answer.
In
bungee jumping, the jumper does not typically suffer injuries similar to
falling directly to the ground because of the elastic cord that decelerates the
jumper gradually, preventing the impact forces that would otherwise cause
severe injury.
When
someone falls, the key factor in determining injury is the sudden stop,
where the body experiences a rapid deceleration as it hits a solid surface,
leading to trauma. In bungee jumping, the elastic cord stretches as the jumper
falls, absorbing much of the kinetic energy over a longer distance and slowing
the fall gradually rather than all at once.
The
critical factors that protect the jumper are:
- Elasticity of the cord: It stretches to slow the
jumper's descent, reducing the force exerted on the body.
- Controlled deceleration: The cord provides a
gradual deceleration, unlike hitting the ground, where the stop is
immediate and severe.
- Proper equipment and setup: The cord length, weight
limits, and jump height are carefully calculated to ensure the jumper
doesn't hit the ground or overstretch the cord.
Injuries
in bungee jumping tend to arise from improper technique, faulty equipment, or
misjudgments in setup, such as using the wrong cord length. Common minor
injuries might include bruising from the harness or stress on the neck or back,
but these are rare when safety precautions are followed.
Since I have studied mathematics and physics 6 decades ago, I decided it is better for me to answer this question using calculations based on Newton’s mechanics and his laws of motion.
Here is how the physics of bungee jumping work
in preventing injury?
Suppose
the jumper weighs 65 kg and he jumps from a height of 200 metres, I asked
myself the kinetic energy, the final velocity, and the momentum at the end of
200 metres jump. If the rope is attached to his waist what would be the
pressure over the waist by the final kinetic energy or the momentum. The
pressure is of course forced over the area which is the waist area. This means
we also need to know a bit on human anatomy, and physiology, not just physics
and mathematics
In
order to calculate the kinetic energy, final velocity, and momentum of the
jumper at the end of the 200-meter fall, we will use some basic physics
equations. Let's start with these calculations.
1.
Kinetic Energy (KE)
At the
end of the 200-meter free fall, all the potential energy (PE) due to gravity is
converted into kinetic energy (ignoring air resistance). The potential energy
is given by:
PE =
mgh
where:
m is
the mass of the jumper (65 kg),
g is
the acceleration due to gravity (9.81 m/s²),
h is
the height (200 m).
Thus,
the potential energy (and hence the kinetic energy at the end of the fall) is:
KE =
mgh = 65×9.81×200 = 127,530 Joules.
2.
Final Velocity (v)
To find
the final velocity, we use the equation derived from the conservation of
energy, where the total potential energy converts into kinetic energy:
v = √
2gh
where,
v =
final velocity
g =
acceleration due to gravity = 9.81 m / s2
h =
height = 200 metres
Substituting
the values:
v = √
(2×9.81×200) = √ 3924 ≈ 62.64 m/s.
So, the
final velocity just before the cord starts stretching would be approximately
62.64 m/s.
The
kinetic energy at the end of the 200-metre fall is given by:
KE = ½
mv2
where,
m =
mass of the jumper = 65 kg
v =
final velocity = 62.64 m /s
Substituting
the values
KE = ½
x 65 kg x (62.64 m/ s)2
KE =
32.5 x (62.64)2 = 127,523 Joules
So, the
kinetic energy at the end of the fall is approximately 127,523 joules (J)
3.
Momentum (p)
Momentum
is given by the formula:
P = mv
Substituting
the values for mass and velocity:
P = 65
× 62.64 = 4,071.6 kg ⋅ m/s.
4.
Pressure on the Waist
To find
the pressure on the jumper’s waist, we need to know the force exerted on the
waist and the area over which this force is distributed.
(i)
Force (F) due to deceleration
If we
assume the bungee cord stops the jumper over a distance d, the force will
depend on how quickly the cord decelerates the jumper. For now, assuming the
deceleration happens over a short distance, we can estimate the force using
Newton's second law:
F
=ma. The force due to momentum would be approximately:
F = Δp
/ Δt
The
actual calculation would depend on how fast the bungee cord stretches and
decelerates the jumper, but for simplicity, we assume the force is significant
and depends largely on the elasticity of the bungee.
(ii)
Waist Area (A)
Let’s
now go a bit into human anatomy. The average waist circumference for an adult
of 65 kg can vary based on body composition, but typically ranges from 70
cm to 85 cm.
Assuming
a circumference of 80 cm (0.8 m), and modelling the waist as a cylinder,
the area is:
A =
waist circumference × waist height ≈ 0.8 × 0.2 = 0.16 m².
Here,
we assume 20 cm as the effective height of the waist region where the force is
distributed.
(iii)
Pressure (P)
The
pressure is the force (F) divided by the area (A):
P = F /
A
Since
the exact deceleration force is complex (dependent on how fast the cord
decelerates), we can calculate this further if given more information about the
bungee cord's elasticity.
Summary of Values
1. Kinetic Energy (KE) = 127,523 Joules
2.Final Velocity (v) = 62.64 m/s.
3.Momentum (p) = 4,071.6 kg·m/s.
4. Waist Area (A) ≈ 0.16 m².
Let’s
jump deeper into the physics of bungee jumping by exploring how the force on
the jumper changes depending on the deceleration distance.
1.
Force During Deceleration
When
the bungee cord starts stretching, it slows the jumper down by applying an
upward force. The force experienced by the jumper depends on how long it takes
(or how far) for the cord to decelerate them to a stop.
To
estimate the force, we can use the work-energy principle:
W = F ×
d = ΔKE
where:
W is
the work done by the force to stop the jumper,
F is
the average force exerted by the cord,
d is
the deceleration distance (how far the cord stretches),
ΔKE is
the change in kinetic energy (which is all the kinetic energy at the point the
cord begins to stretch).
Since
we know the kinetic energy (ΔKE =127,530 J), we can rearrange the
equation to find the force:
F = ΔKE
/ d
Now,
let’s compute the force for different deceleration distances.
2.
Force Calculation for Different Deceleration Distances
(i)
Deceleration Distance = 10 meters
If the
cord stretches by 10 meters to decelerate the jumper, we can calculate the
force as:
F =
127,530 J / 10 m = 12,753 Newton
This
means the force exerted on the jumper’s waist would be about 12,753
Newtons.
(ii)
Deceleration Distance = 20 meters
If the
cord stretches over 20 meters:
F =
127,530 J / 20 m = 6,376.5 N
Here,
the force decreases to 6,376.5 Newtons, since a longer deceleration
distance means a gentler stop.
(iii)
Deceleration Distance = 30 meters
If the
cord stretches over 30 meters:
F =
127,530 J / 30 m = 4,251 N
The
force drops to 4,251 Newtons over 30 meters, making the stop even
gentler.
Interesting
Observations on the Physics of Bungee Jumping
a) Elasticity
and Safety: The elasticity of the bungee cord is key to safety. A
cord that stretches more can decelerate the jumper over a longer distance,
reducing the force on the body and preventing injury. A shorter stretch would
lead to a harsher stop and a higher risk of injury.
b) Acceleration
Forces (g-forces): The force felt by the jumper is often expressed in terms
of g-forces. One "g" is the force of gravity (9.81 m/s²) that we
experience on Earth. When bungee jumping, the force during deceleration can be
several times the normal gravitational force. For example:
g-force
= F / mg
For a
force of 12,753 N (in the 10-meter deceleration case):
g-force
= (12,753 / 65×9.81) ≈ 20 g
This
means the jumper would experience about 20 times the force of gravity during
deceleration, which is extreme but manageable in short bursts for a fit person.
A longer deceleration distance reduces the g-forces and makes the experience
less intense.
a) Air
Resistance: While we’ve neglected air resistance in our calculations, it would
play a role in a real jump by reducing the final velocity slightly before the
cord begins to stretch. This effect would be more noticeable on very high
jumps.
b) Waist
Pressure: To calculate the pressure over the waist for these forces, we use the
pressure formula
P = F /
A. For instance, with the force of 12,753 N over an area
of 0.16 m² (as estimated earlier):
P =
12,753 / 0.16 = 79,706.25 N/m² (or Pascals)
This
pressure would be quite significant, emphasizing the importance of a
well-designed, padded harness to distribute the load over a larger area and
prevent injury to the jumper's body.
4.
Energy Dissipation in the Bungee Cord
The bungee
cord dissipates the energy of the fall by converting the jumper’s kinetic
energy into elastic potential energy. If the cord is too stiff, it won’t
stretch enough, and the forces on the jumper will be too large, leading to
injury. If it’s too elastic, the jumper might come too close to the ground.
Final
Thoughts:
We
started off using logical assumptions without mathematical evidence. We then
used physics to prove our assumption.
The
physics of bungee jumping is a beautiful interplay between energy conservation
and elasticity. The elastic cord ensures that the energy from the fall is
absorbed over time and distance, which minimizes the force on the jumper.
Proper equipment and setup are critical in managing the forces and ensuring a
safe and thrilling experience.
Having
discussed and explained all the physics of this jump from a height, would you
like to try?
Also
see this article here:
What
Happens When a Human Falls from 10,000 metres?
https://scientificlogic.blogspot.com/search?q=falling+from+a+height
Jb lim
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