Tarzan is running towards a cliff, ready to swing down rescue Jane from some poisonous snakes, and make it back
up to a tree on the other side. For now, we are going to assume energy is conserved and ignore momentum. If the height of the cliff is 10m, the height of the tree on which Tarzan hopes to land is 7 m, and the mass of Tarzan is 100kg and the mass of Jane is 50 kg. What velocity does Tarzan have to be running to rescue Jane?
3.13
To find the velocity at which Tarzan needs to be running in order to rescue Jane, we can use the principle of conservation of energy. Conservation of energy states that the initial energy of a system is equal to the final energy of the system.
In this case, the initial energy of the system is the kinetic energy of Tarzan when he is running towards the cliff, and the final energy is the potential energy of Tarzan and Jane when they are at their respective positions.
The kinetic energy of Tarzan is given by the formula:
KE = (1/2) * Mass * Velocity^2
The potential energy of Tarzan when he is at a height of 10m (before swinging down) is given by:
PE_initial = Mass * gravity * Height_initial
The potential energy of Tarzan and Jane when they are at a height of 7m (when Tarzan hopes to land on the tree) is given by:
PE_final = (Mass_of_Tarzan + Mass_of_Jane) * gravity * Height_final
According to the principle of conservation of energy, the initial energy (kinetic energy) should be equal to the final energy (potential energy). So we can set up the equation:
(1/2) * Mass * Velocity^2 = (Mass_of_Tarzan + Mass_of_Jane) * gravity * Height_final - Mass * gravity * Height_initial
Plugging in the values:
- Mass_of_Tarzan = 100kg
- Mass_of_Jane = 50kg
- Height_initial = 10m
- Height_final = 7m
- gravity = 9.8 m/s^2 (acceleration due to gravity)
The equation becomes:
(1/2) * 100 * Velocity^2 = (100 + 50) * 9.8 * 7 - 100 * 9.8 * 10
Simplifying:
50 * Velocity^2 = 150 * 9.8 * 7 - 100 * 9.8 * 10
Velocity^2 = (150 * 9.8 * 7 - 100 * 9.8 * 10) / 50
Calculating:
Velocity^2 = 882 - 490
Velocity^2 = 392
Taking the square root of both sides:
Velocity = sqrt(392)
Calculating the value of Velocity:
Velocity ≈ 19.8 m/s
Therefore, Tarzan needs to be running at a velocity of approximately 19.8 m/s to rescue Jane.