A 72 Kg Stunt Woman Jumps From A 29 M High Roof And Lands On A Ledge 18 M High. What Is Her Change In
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Introduction
Stunt performances often involve precise calculations of physics principles, particularly energy transformations and motion dynamics, to ensure safety and realism. When a stunt woman jumps from a significant height and lands on a ledge at a lower elevation, understanding the change in her physical state requires analyzing various factors like gravitational potential energy, kinetic energy, and the effects of impact forces. This article provides an in-depth exploration of her change in energy, velocity, and momentum during this daring stunt, with step-by-step calculations and explanations.
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Understanding the Scenario
Basic Details
- Mass of the stunt woman (m): 72 kg
- Initial height (h₁): 29 meters (height of the roof)
- Ledge height (h₂): 18 meters
- Difference in height (Δh): 11 meters
- Gravity (g): 9.81 m/s²
The stunt involves her jumping from the roof, descending under gravity, and landing safely on the ledge. During this process, her energy states change significantly, from potential energy at the start to kinetic energy just before landing, and then dissipate upon impact.
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Calculating Her Initial Potential Energy
Potential Energy at the Starting Point
Potential energy (PE) is defined as:
\[ PE = m \times g \times h \]
At her initial position (height h₁):
\[ PE_{initial} = 72\,kg \times 9.81\,m/s^2 \times 29\,m \]
Calculating:
\[ PE_{initial} = 72 \times 9.81 \times 29 \]
\[ PE_{initial} \approx 72 \times 284.49 \]
\[ PE_{initial} \approx 20,507.28\,J \]
This energy represents her maximum potential energy before the jump.
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Calculating Her Potential Energy at the Landing Point
At the landing ledge (height h₂ = 18 m):
\[ PE_{final} = 72\,kg \times 9.81\,m/s^2 \times 18\,m \]
Calculating:
\[ PE_{final} = 72 \times 9.81 \times 18 \]
\[ PE_{final} \approx 72 \times 176.58 \]
\[ PE_{final} \approx 12,727.76\,J \]
The reduction in potential energy during her fall is:
\[ \Delta PE = PE{initial} - PE{final} \]
\[ \Delta PE \approx 20,507.28\,J - 12,727.76\,J \]
\[ \Delta PE \approx 7,779.52\,J \]
This energy difference is primarily converted into kinetic energy and energy dissipated due to air resistance and impact forces.
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Her Velocity at the Moment of Landing
Theoretical Velocity Assuming No Air Resistance
Ignoring air resistance, the velocity of her just before landing can be found using energy conservation:
\[ KE = PE{initial} - PE{final} \]
\[ KE = 7,779.52\,J \]
Kinetic energy (KE) is also:
\[ KE = \frac{1}{2} m v^2 \]
Solving for \( v \):
\[ v = \sqrt{\frac{2 \times KE}{m}} \]
\[ v = \sqrt{\frac{2 \times 7,779.52}{72}} \]
\[ v = \sqrt{\frac{15,559.04}{72}} \]
\[ v = \sqrt{215.82} \]
\[ v \approx 14.7\,m/s \]
Thus, her velocity just before impact is approximately 14.7 m/s.
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Impact and Energy Dissipation
Real-World Considerations
In reality, some energy is lost due to air resistance and deformation during impact, meaning her actual impact velocity might be slightly lower or higher depending on factors like her posture, air drag, and the surface properties.
Change in Her Energy State
- Initial energy: 20,507.28 J (potential energy at 29 m)
- Energy at impact (kinetic): approximately 14.7 m/s velocity corresponding to KE of 7,779.52 J
- Energy after impact: some energy is dissipated as heat, sound, and deformation of her body and the ledge
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Momentum Analysis During the Jump
Momentum at Impact
Momentum (p) is:
\[ p = m \times v \]
Using her impact velocity:
\[ p = 72\,kg \times 14.7\,m/s \]
\[ p \approx 1,058.4\,kg \cdot m/s \]
This momentum reflects her state just before impact and is crucial for understanding force and safety measures.
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The Change in Her Physical State: Energy and Momentum Perspective
Summary of Changes
| Quantity | Initial | Final (Impact) | Change |
|---|---|---|---|
| Potential Energy | 20,507.28 J | ~12,727.76 J (at ledge) | Decrease of ~7,779.52 J |
| Kinetic Energy | 0 | ~7,779.52 J | Increase of ~7,779.52 J |
| Velocity | 0 (initial at top) | ~14.7 m/s | Increase of ~14.7 m/s |
| Momentum | 0 | ~1,058.4 kg·m/s | Increase accordingly |
Her energy transforms from potential to kinetic during the fall, with impact forces dissipating some energy to prevent injury.
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Safety and Practical Implications
Protective Measures
- Harnesses and safety nets: To absorb impact forces and reduce the risk of injury.
- Controlled environment: Stunt performers practice with precise calculations to ensure safe landings.
- Body positioning: Techniques like bending knees reduce the shock transmitted to the body.
Physics in Stunt Planning
Understanding the change in energy and velocity helps stunt coordinators design safer sequences, ensuring that impact forces are manageable and injuries minimized.
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Conclusion
The daring jump of a 72 kg stunt woman from a 29-meter-high roof onto an 18-meter-high ledge exemplifies a practical application of physics principles. Her potential energy at the start (~20,507 J) decreases significantly during her fall, converting into kinetic energy (~7,779.52 J) as she approaches the ledge. Her impact velocity reaches approximately 14.7 m/s, with a momentum of about 1,058 kg·m/s. These calculations highlight the importance of energy transformation understanding in stunt safety and execution.
While real-world stunts involve numerous safety measures to mitigate risks, the physics underpinning her motion provides essential insights into the forces and energies at play during such high-stakes performances.
