How Is Potential Energy Converted to Kinetic Energy in Physical Systems?

  • #1
Sidhshah1234
1
0
Homework Statement
A daredevil gazelle called Giselle takes her first bungee jump from a height of 160 m above the ground. Giselle’s mass is 26 kg, and when she is attached by the ankles to the bungee cord, her height means she will extend 2.0 m beyond the end of the bungee cord. Her bungee cord is 33 m long.

(a) How far does Giselle travel downward in freefall before the spring effect of the bungee stretching kicks in?[Ans: 33 m]

(b) Draw the free-body diagram of Giselle when the bungee spring force is acting.

(c) What was Giselle’s initial potential energy as she stepped off the platform at the top? [Ans: 41,000 J]

(d) Giselle’s bungee instructor has carefully chosen her bungee cord accounting for the platform height and Giselle’s weight, in order that she will just barely touch the ground. What is the spring constant, k, of the bungee cord?[Ans: 5.2 N/m]

(e) Recognizing that Giselle’s maximum velocity occurs when her acceleration is zero (she is at the point between speeding up and the bungee cord slowing her down again), how far has she fallen when she reaches this maximum velocity? [Ans: 82 m]

(f) What is her maximum velocity? [Ans: 34 ms-1]

I need help with d) - f)
Relevant Equations
F=kx
F=ma
PE=mgh
W=1/2kx^2
d)
W=1/2kx^2=PE
41000=1/2(k)(33)^2
k=82000/(33)^2
k=75.3N/m
(nowhere close to the answer)

e)
Potential Energy = Kinetic Energy
mgh = 0.5 * m * v^2

Given:
m = 26 kg
g = 9.8 m/s^2
h = 160 m

Solving for velocity (v):

0.5 * 26 kg * v^2 = 26 kg * 9.8 m/s^2 * 160 m
v^2 = (2 * 9.8 m/s^2 * 160 m)
v = sqrt(2 * 9.8 m/s^2 * 160 m) ≈ 56 m/s
v^2 = u^2 + 2as
0 = 0^2 + 2 * 9.8 m/s^2 * ss = 0.5 * (34 m/s)^2 / 9.8 m/s^2 ≈ 160m
 
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  • #2
Hello and :welcome: !
Sidhshah1234 said:
(d) Giselle’s bungee instructor has carefully chosen her bungee cord accounting for the platform height and Giselle’s weight, in order that she will just barely touch the ground. What is the spring constant, k, of the bungee cord?[Ans: 5.2 N/m]


I need help with d) - f)
Relevant Equations: F=kx
F=ma
PE=mgh
W=1/2kx^2

41000=1/2(k)(33)^2
Did you make a sketch ? Please post it ...

What is this ##x=33 ## on the sketch ?

##\ ##
 
  • #3
For the record:
  • what value of ##g## are you supposed to use ? Ah, I see a 9.8 m/s2 in your part e) answer (*)
  • I have trouble with the book answer for part a). If
    she stepped off the platform at the top
    and the 33 m answer is correct she is still upright !?
  • The problem statement gives you the girl's length. How do you use that ? (my answer: you seem to ignore it. But: What if she was 20 m high instead of 2 ? )
(*):
Sidhshah1234 said:
e)
Potential Energy = Kinetic Energy
mgh = 0.5 * m * v^2

Given:
m = 26 kg
g = 9.8 m/s^2
h = 160 m

Solving for velocity (v):

0.5 * 26 kg * v^2 = 26 kg * 9.8 m/s^2 * 160 m
v^2 = (2 * 9.8 m/s^2 * 160 m)
v = sqrt(2 * 9.8 m/s^2 * 160 m) ≈ 56 m/s
v^2 = u^2 + 2as
0 = 0^2 + 2 * 9.8 m/s^2 * ss = 0.5 * (34 m/s)^2 / 9.8 m/s^2 ≈ 160m
This looks a lot like a disaster where someone forgot the bungee cord :smile:


##\ ##
 
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  • #4
Sidhshah1234 said:
W=1/2kx^2=PE
41000=1/2(k)(33)^2
You are saying the bungee cord stretches 33 m?
 
  • #5
Sidhshah1234 said:
Homework Statement: A daredevil gazelle called Giselle takes her first bungee jump from a height of 160 m above the ground. Giselle’s mass is 26 kg, and when she is attached by the ankles to the bungee cord, her height means she will extend 2.0 m beyond the end of the bungee cord. Her bungee cord is 33 m long.

(a) How far does Giselle travel downward in freefall before the spring effect of the bungee stretching kicks in?[Ans: 33 m]
What about Giselle's height? The problem setter wants us to consider it.

As I see it, Giselle is about 2 meters in height. Her center of gravity begins about 1 meter above her ankles (161 meters above the ground). During the head-first fall, her center of gravity is about 1 meter below her ankles.
 
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Likes erobz
  • #6
The question setter apparently also thinks it’s a good idea that the bungee operator chooses a bungee such that her head just barely touches the ground in part d…🫣
 
  • #7
BvU said:
and the 33 m answer is correct she is still upright !?
She could well be still upright at the instant the cord becomes taut. Her orientation would change abruptly after that.
 

FAQ: How Is Potential Energy Converted to Kinetic Energy in Physical Systems?

What is potential energy and how is it stored in physical systems?

Potential energy is the energy possessed by an object due to its position, condition, or configuration. It is stored in physical systems in various forms, such as gravitational potential energy (due to an object's height above the ground), elastic potential energy (stored in stretched or compressed springs), and chemical potential energy (stored in chemical bonds).

How does potential energy convert to kinetic energy?

Potential energy converts to kinetic energy when the position or condition of an object changes, causing it to move. For example, when an object falls from a height, its gravitational potential energy is converted into kinetic energy as it accelerates towards the ground. Similarly, a compressed spring releases its elastic potential energy as kinetic energy when it is allowed to expand.

What is the role of conservation of energy in the conversion process?

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. During the conversion of potential energy to kinetic energy, the total energy of the system remains constant. As potential energy decreases, an equivalent amount of kinetic energy increases, ensuring the total energy is conserved.

Can you provide an example of potential energy converting to kinetic energy in everyday life?

A common example is a roller coaster. At the highest point of the track, the coaster has maximum gravitational potential energy. As it descends, this potential energy is converted into kinetic energy, increasing its speed. When it climbs up another hill, kinetic energy is converted back into potential energy.

What factors affect the efficiency of potential energy conversion to kinetic energy?

The efficiency of potential energy conversion to kinetic energy can be affected by factors such as friction, air resistance, and mechanical losses. In an ideal, frictionless system, all potential energy would convert to kinetic energy. However, in real-world systems, some energy is always lost to heat, sound, or other forms of energy, reducing the efficiency of the conversion.

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