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QuestionThe 76.2-kg Tarzan uses a 4.17-m vine to swing from a tree to rescue 53.4-kg Jane whose life is endangered by a poisonous snake. The vine starts horizontal when Tarzan begins his rescue. After rescuing Jane, the two continue to swing in a pendulum-like motion. To what maximum height will the two swing before losing all their kinetic energy?

Studdy Solution

STEP 1

1. Tarzan and Jane are treated as point masses.
2. The vine is massless and inextensible.
3. The system is isolated, and energy is conserved.
4. Air resistance and other non-conservative forces are negligible.
5. The vine starts from a horizontal position.

STEP 2

1. Calculate the initial potential energy of Tarzan.
2. Determine the total mass after Tarzan picks up Jane.
3. Use conservation of energy to find the maximum height.

STEP 3

Calculate the initial potential energy of Tarzan when the vine is horizontal:
The potential energy PE PE is given by:
PE=m⋅g⋅h PE = m \cdot g \cdot h
where: - m=76.2 kg m = 76.2 \, \text{kg} (mass of Tarzan), - g=9.81 m/s2 g = 9.81 \, \text{m/s}^2 (acceleration due to gravity), - h=4.17 m h = 4.17 \, \text{m} (height of the vine when horizontal).
PEinitial=76.2 kg×9.81 m/s2×4.17 m PE_{\text{initial}} = 76.2 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 4.17 \, \text{m}

STEP 4

Determine the total mass after Tarzan picks up Jane:
mtotal=mTarzan+mJane m_{\text{total}} = m_{\text{Tarzan}} + m_{\text{Jane}} =76.2 kg+53.4 kg = 76.2 \, \text{kg} + 53.4 \, \text{kg} =129.6 kg = 129.6 \, \text{kg}

STEP 5

Use conservation of energy to find the maximum height:
The initial potential energy of Tarzan is converted into the potential energy of Tarzan and Jane at the maximum height:
PEinitial=PEfinal PE_{\text{initial}} = PE_{\text{final}}
mTarzan⋅g⋅hinitial=mtotal⋅g⋅hfinal m_{\text{Tarzan}} \cdot g \cdot h_{\text{initial}} = m_{\text{total}} \cdot g \cdot h_{\text{final}}
Cancel g g from both sides:
mTarzan⋅hinitial=mtotal⋅hfinal m_{\text{Tarzan}} \cdot h_{\text{initial}} = m_{\text{total}} \cdot h_{\text{final}}
Solve for hfinal h_{\text{final}} :
hfinal=mTarzan⋅hinitialmtotal h_{\text{final}} = \frac{m_{\text{Tarzan}} \cdot h_{\text{initial}}}{m_{\text{total}}}
Substitute the known values:
hfinal=76.2 kg×4.17 m129.6 kg h_{\text{final}} = \frac{76.2 \, \text{kg} \times 4.17 \, \text{m}}{129.6 \, \text{kg}}
Calculate hfinal h_{\text{final}} :
hfinal≈317.754129.6≈2.45 m h_{\text{final}} \approx \frac{317.754}{129.6} \approx 2.45 \, \text{m}
The maximum height the two will swing to is:
2.45 m \boxed{2.45 \, \text{m}}

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