Tarzan, King of the Jungle (mass = 73.2 kg), grabs a vine of length 14.1 m hanging from a tree branch. The angle of the vine was 25.9° with respect to the vertical when he grabbed it. At the lowest point of his trajectory, he picks up Jane (mass = 41.0 kg) and continues his swinging motion. What angle relative to the vertical will the vine have when Tarzan and Jane reach the highest point of their trajectory?
The grabbing of Jane at the lowest point of the swing is a momentum-conserving event, but some kinetic energy is lost. Then they swing to the highest point with energy conserved again.
Let Tarzan's speed before grabbing Jane = V1:
Tarzan's mass = Mt
Jane's mass = Mj
Mt*g*(1-cos25.9) = (Mt/2)*V1^2
V1 = sqrt[2*g*(1-cos25.9)] = 1.403 m/s
After grabbing Jane,
V1*Mt = V2(Mt+Mj)
V2 = V1*[Mt/(Mt+Mj)]= 0.641V1 = 0.899 m/s
Final maximum angle A is given by
g(1-cosA) = (1/2)V2^2
1 - cosA = 0.04126
cosA = 0.9587
A = 16.5 degrees
To find the angle relative to the vertical when Tarzan and Jane reach the highest point of their trajectory, we can use the principle of conservation of mechanical energy.
First, let's find the initial potential energy of Tarzan when he grabs the vine. Tarzan's mass is 73.2 kg, and the gravitational acceleration is approximately 9.8 m/s^2. So, his initial potential energy is given by:
Potential Energy = Mass * gravity * height
= 73.2 kg * 9.8 m/s^2 * (14.1 m * sin(25.9°))
Next, let's consider the highest point of their trajectory. At this point, all of Tarzan and Jane's initial potential energy is converted into gravitational potential energy and kinetic energy. Since we know the total mechanical energy is conserved, we can equate the initial potential energy to the sum of their gravitational potential energy and kinetic energy at the highest point.
The gravitational potential energy is given by:
Gravitational Potential Energy = (Tarzan's mass + Jane's mass) * gravity * height
= (73.2 kg + 41.0 kg) * 9.8 m/s^2 * (14.1 m)
The kinetic energy at the highest point is zero since they momentarily stop. The total mechanical energy equation becomes:
Potential Energy = Gravitational Potential Energy
Simplifying the equation, we get:
73.2 kg * 9.8 m/s^2 * (14.1 m * sin(25.9°)) = (73.2 kg + 41.0 kg) * 9.8 m/s^2 * (14.1 m)
Now, we can solve this equation to find the angle relative to the vertical when Tarzan and Jane reach the highest point.