Tarzan, whose mass is 76 kg, travelling at 9.5m/s, grabs from the bottom of a 1.5 m long vine and swings upward.

a) Calculate the maximum height of Tarzan.
b) Calculate the angle at which the vine makes with the vertical at his maximum height.

find his KE as he grabs on. Now convert that to PE.

mgh=1/2 mv^2
solve for h.

angle:
draw the figure.
arccos=(1.50-height)/1.5

To solve this problem, we need to consider the conservation of energy and the principles of projectile motion. Let's break it down step-by-step:

a) To calculate the maximum height of Tarzan, we need to find the potential energy Tarzan has gained when he reaches the highest point of his swing. At that point, all of his initial kinetic energy will have been converted into potential energy.

First, let's find Tarzan's initial kinetic energy (KEi):
KEi = (1/2) * mass * velocity^2
KEi = (1/2) * 76 kg * (9.5 m/s)^2
KEi = 34220 J

Next, we use the conservation of energy to calculate his maximum potential energy (PEmax). Since there is no energy lost to external forces like air resistance, the total initial kinetic energy will be equal to the maximum potential energy at the highest point:
PEmax = KEi

To find the height at the highest point, we can use the formula for potential energy:
PE = m * g * h
Where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.

Therefore, at the maximum height (hmax):
PEmax = m * g * hmax
hmax = PEmax / (m * g)
hmax = 34220 J / (76 kg * 9.8 m/s^2)
hmax ≈ 45.54 m

So, the maximum height Tarzan reaches is approximately 45.54 meters.

b) To find the angle at which the vine makes with the vertical at Tarzan's maximum height, we can use trigonometric properties.

Let's call the angle θ. At the highest point, the velocity will be entirely horizontal. Therefore, the vertical component of velocity (Vy) will be zero.

Using the equation for vertical velocity at the highest point:
Vy = velocity * sin(θ) = 0

Since Vy = 0, we know that sin(θ) = 0 because the sine of 0 degrees is 0. Therefore, θ = 0 degrees.

Hence, the angle at which the vine makes with the vertical at Tarzan's maximum height is 0 degrees.