The initial potential energy of the wagon containing gold boxes will enable
it roll down the hill when cut loose.
The Lone Ranger and Tonto have approximately <u>5.1 seconds</u>.
Reasons:
Mass wagon and gold = 166 kg
Location of the wagon = 77 meters up the hill
Slope of the hill = 8°
Location of the rangers = 41 meters from the canyon
Mass of Lone Ranger, m₁ = 65 kg
Mass of Tonto m₂ = 66 kg
Solution;
Height of the wagon above the level ground, h = 77 m × sin(8°) ≈ 10.72 m
Potential energy = m·g·h
Where;
g = Acceleration due to gravity ≈ 9.81 m/s²
Potential energy of wagon, P.E. ≈ 166 × 9.81 × 10.72 = 17457.0912
Potential energy of wagon, P.E. ≈ 17457.0912 J
By energy conservation, P.E. = K.E.
![K.E. = \mathbf{\dfrac{1}{2} \cdot m \cdot v^2}](https://tex.z-dn.net/?f=K.E.%20%3D%20%5Cmathbf%7B%5Cdfrac%7B1%7D%7B2%7D%20%5Ccdot%20m%20%5Ccdot%20v%5E2%7D)
Where;
v = The velocity of the wagon a the bottom of the cliff
Therefore;
![\dfrac{1}{2} \times 166 \times v^2 = 17457.0912](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B2%7D%20%5Ctimes%20166%20%5Ctimes%20v%5E2%20%3D%2017457.0912)
![v = \sqrt{\dfrac{17457.0912}{\dfrac{1}{2} \times 166} } \approx 14.5](https://tex.z-dn.net/?f=v%20%3D%20%5Csqrt%7B%5Cdfrac%7B17457.0912%7D%7B%5Cdfrac%7B1%7D%7B2%7D%20%5Ctimes%20166%7D%20%7D%20%5Capprox%2014.5)
Velocity of the wagon, v ≈ 14.5 m/s
Momentum = Mass, m × Velocity, v
Initial momentum of wagon = m·v
Final momentum of wagon and ranger = (m + m₁ + m₂)·v'
By conservation of momentum, we have;
m·v = (m + m₁ + m₂)·v'
![\therefore v' = \mathbf{ \dfrac{m \cdot v}{(m + m_1 + m_2) }}](https://tex.z-dn.net/?f=%5Ctherefore%20v%27%20%3D%20%5Cmathbf%7B%20%5Cdfrac%7Bm%20%5Ccdot%20v%7D%7B%28m%20%2B%20m_1%20%2B%20m_2%29%20%20%7D%7D)
Which gives;
![\therefore v' = \dfrac{166 \times 14.5}{(166 + 65 + 66) } \approx 8.1](https://tex.z-dn.net/?f=%5Ctherefore%20v%27%20%3D%20%5Cdfrac%7B166%20%5Ctimes%2014.5%7D%7B%28166%20%2B%2065%20%2B%2066%29%20%20%7D%20%5Capprox%208.1)
The velocity of the wagon after the Ranger and Tonto drop in, v' ≈ 8.1 m/s
![Time = \dfrac{Distance}{Velocity}](https://tex.z-dn.net/?f=Time%20%3D%20%5Cdfrac%7BDistance%7D%7BVelocity%7D)
![\mathrm{The \ time \ the\ Lone \ Ranger \ and \ Tonto \ have, \ t} = \dfrac{41 \, m}{8.1 \, m/s} \approx 5.1 \, s](https://tex.z-dn.net/?f=%5Cmathrm%7BThe%20%5C%20time%20%5C%20the%5C%20Lone%20%5C%20%20Ranger%20%5C%20%20and%20%20%5C%20Tonto%20%5C%20%20have%2C%20%20%5C%20t%7D%20%3D%20%5Cdfrac%7B41%20%5C%2C%20m%7D%7B8.1%20%5C%2C%20m%2Fs%7D%20%5Capprox%205.1%20%5C%2C%20s)
The Lone Range and Tonto have approximately <u>5.1 seconds</u> to grab the
gold and jump out of the wagon before the wagon heads over the cliff.
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