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?
To find the velocity at which Tarzan needs to be running to rescue Jane, we can use the principle of conservation of energy.
First, let's analyze the energy changes that occur during this scenario:
1. Tarzan's initial kinetic energy (KE1) while he is running.
2. Tarzan's potential energy (PE1) when he is at the top of the cliff.
3. Tarzan's kinetic energy (KE2) when he swings down to rescue Jane.
4. Jane's potential energy (PE2) when Tarzan rescues her.
5. Tarzan's potential energy (PE3) when he reaches the tree on the other side.
According to the conservation of energy principle, the total mechanical energy at any point should remain constant. Therefore, we can write:
KE1 + PE1 = KE2 + PE2 + PE3
Let's calculate these energies one by one:
1. Tarzan's initial kinetic energy (KE1):
KE1 = (1/2) * mass * velocity^2
Given that Tarzan's mass is 100 kg and we need to find velocity, we'll keep it as a variable.
2. Tarzan's potential energy (PE1):
PE1 = mass * gravitational acceleration * height
Given that the height of the cliff is 10 m, the mass of Tarzan is 100 kg, and the gravitational acceleration is approximately 9.8 m/s^2.
3. Tarzan's kinetic energy (KE2):
KE2 = (1/2) * mass * velocity^2
Again, we need to keep it as a variable.
4. Jane's potential energy (PE2):
PE2 = mass * gravitational acceleration * height
Given that the height of the tree is 7 m and Jane's mass is 50 kg.
5. Tarzan's potential energy (PE3):
PE3 = mass * gravitational acceleration * height
Given that the height of the tree across the cliff is also 7 m.
Now, let's set up the equation:
(1/2) * 100 * velocity^2 + 100 * 9.8 * 10 = (1/2) * 100 * velocity^2 + 50 * 9.8 * 7 + 100 * 9.8 * 7
Simplifying the equation:
4900 = 1715 + 3430 + 6860 - (1/2) * 100 * velocity^2
Rearranging the equation and grouping the terms:
(1/2) * 100 * velocity^2 = 1715 + 3430 + 6860 - 4900
(1/2) * 100 * velocity^2 = 7105
Dividing both sides by 50:
100 * velocity^2 = 142100
Dividing both sides by 100:
velocity^2 = 1421
Taking the square root of both sides:
velocity = √1421
Therefore, the velocity at which Tarzan needs to be running to rescue Jane is approximately 37.68 m/s.
To calculate the velocity at which Tarzan needs to be running to rescue Jane, we can use conservation of energy.
The potential energy of the system at the top of the cliff is converted into kinetic energy as Tarzan swings down to rescue Jane and then back up to the tree.
The potential energy at the top of the cliff is given by:
Potential Energy = mass * gravitational acceleration * height
Since Tarzan's mass is 100 kg and the height of the cliff is 10 m, we have:
Potential Energy at the top = 100 kg * 9.8 m/s^2 * 10 m = 9800 J
The potential energy at the top is converted into kinetic energy at the bottom of the swing. At the bottom, there is no potential energy and all of the initial potential energy is converted into kinetic energy.
So, we have:
Potential Energy at the bottom = 0 J
The kinetic energy at the bottom is given by:
Kinetic Energy at the bottom = (1/2) * mass * velocity^2
Since Tarzan's mass is 100 kg, we have:
Kinetic Energy at the bottom = (1/2) * 100 kg * velocity^2
Since energy is conserved, the potential energy at the top is equal to the kinetic energy at the bottom:
Potential Energy at the top = Kinetic Energy at the bottom
Therefore, we can equate the two:
9800 J = (1/2) * 100 kg * velocity^2
Simplifying the equation, we have:
9800 J = 50 kg * velocity^2
Dividing both sides by 50 kg, we get:
196 = velocity^2
Taking the square root of both sides, we have:
velocity = √196
Therefore, the velocity at which Tarzan needs to be running to rescue Jane is 14 m/s.