Tarzan, played by Patrick Huard, 60 kg, must cross a stream

5 m wide, and full of hungry crocodiles. To do this,
decides to slide along a sloping tree trunk, hollowed
and moist so that there is no friction. After the trunk,
there is a part of 3 m platform covered in mud, with a
coefficient of kinetic friction of 0.3 leading to the edge of
stream. It clings so, without jumping to a vine rigid
8 m with a massless, free to rotate (without friction), in
lower end which is just below where it is
attached (the vine hangs). The vine and he traveled an arc
above the stream. Tarzan must release the vine when
stops momentarily and land, dry, on the other side of
stream!
a) You must ensure that nothing bad will happen to
Tarzan, and for this you must select the tree trunk to be used (starting height of Tarzan).
b) The actor who plays the role of Tarzan wants to land on a chair mounted on a spring so as not to injure,
is stipulated in his contract. This chair is 1 m above the ground and is attached to a spring
can undergo a maximum compression of 0.3 m. What should be the spring constant of the spring to avoid
not damage the player?

To solve this problem, we need to consider the different stages of Tarzan's journey and apply the relevant principles of physics. Let's break it down step by step:

a) Selecting the tree trunk:
Tarzan needs to select a tree trunk with a specific starting height to ensure he can safely cross the stream. First, let's consider Tarzan's initial potential energy (PE) and his final kinetic energy (KE) after sliding down the tree trunk.

1. Tarzan's initial potential energy (PE):
PE = m * g * h
Where m is Tarzan's mass (60 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the tree trunk.

2. Tarzan's final kinetic energy (KE):
KE = (1/2) * m * v^2
Where v is Tarzan's final velocity after sliding down the tree trunk.

3. Considering conservation of energy:
Since there is no friction while sliding down the tree trunk, Tarzan's initial potential energy should be equal to his final kinetic energy:
PE = KE

Solving for Tarzan's starting height (h):
m * g * h = (1/2) * m * v^2
h = (1/2) * v^2 / g

b) The spring constant:
Tarzan wants to land on a chair mounted on a spring to avoid injury. To ensure the spring can support his weight and absorb the impact, we need to calculate the required spring constant (k). The potential energy stored in the compressed spring should equal Tarzan's potential energy just before landing.

1. Tarzan's potential energy just before landing:
PE = m * g * h
Where h is the height from the ground to the top of the compressed spring (1 m + 0.3 m).

2. Potential energy stored in the compressed spring:
PE = (1/2) * k * x^2
Where k is the spring constant, and x is the maximum compression of the spring (0.3 m).

Solving for the spring constant (k):
m * g * h = (1/2) * k * x^2
k = 2 * m * g * h / x^2

Now you can plug in the values and calculate both the starting height of the tree trunk (from a) and the required spring constant (from b).