Tarzan plans to cross a gorge by swinging in an arc from a hanging vine. If his arms are capable of exerting a force of 1820 N on the rope, what is the maximum speed (in meters/second) he can tolerate at the lowest point of his swing? His mass is 83.0 kg, and the vine is 11.0 m long.
M*(g + V^2/R) = 1820 N
at maximum allowed rope tension.
Solve for V.
To determine the maximum speed Tarzan can tolerate at the lowest point of his swing, we can use the principle of conservation of mechanical energy. At the highest point of the swing, all of Tarzan's potential energy is converted into kinetic energy. At the lowest point, all of his kinetic energy is converted into potential energy.
1. Calculate Tarzan's potential energy at the highest point:
Potential energy = mass * gravity * height
Tarzan's mass = 83.0 kg
Acceleration due to gravity (g) = 9.8 m/s²
Height = length of the vine = 11.0 m
Potential energy at the highest point = 83.0 kg * 9.8 m/s² * 11.0 m = 9,284.6 J
2. Calculate Tarzan's maximum kinetic energy at the lowest point:
Maximum kinetic energy = Potential energy at highest point
Kinetic energy = 0.5 * mass * velocity²
Tarzan's mass = 83.0 kg
Maximum kinetic energy = 9,284.6 J
Substitute the values into the equation:
0.5 * 83.0 kg * velocity² = 9,284.6 J
3. Rearrange the equation to solve for velocity:
velocity² = (2 * Maximum kinetic energy) / mass
velocity² = (2 * 9,284.6 J) / 83.0 kg
velocity² = 223.854 J/kg
4. Take the square root of both sides to find the velocity:
velocity = √(223.854 J/kg)
Calculating the value:
velocity = 14.973 m/s (rounded to three decimal places)
Therefore, the maximum speed that Tarzan can tolerate at the lowest point of his swing is approximately 14.973 meters per second.