Tarzan swings on a 28.0 m long vine initially inclined at an angle of 22° from the vertical.

(a) What is his speed at the bottom of the swing if he starts from rest?
(b) What is his speed at the bottom of the swing if he starts with an initial speed of 3.06 m/s?

(a) Well, Tarzan certainly knows how to make an entrance! To find his speed at the bottom of the swing when starting from rest, we can use the concept of conservation of mechanical energy. At the top of the swing, all of Tarzan's initial potential energy is converted into kinetic energy at the bottom of the swing.

So, let's calculate! The initial potential energy is given by m*g*h, where m is Tarzan's mass, g is the acceleration due to gravity (roughly 9.8 m/s^2), and h is the vertical height at the top of the swing.

To find h, we can use the given length of the vine and the angle of inclination. The vertical height can be calculated as h = l * sin(θ), where l is the length of the vine and θ is the angle of inclination.

So, h = 28.0 m * sin(22°) = 10.23 m (approximately).

Now, the initial potential energy is m * g * h = m * 9.8 m/s^2 * 10.23 m.

Since all of the potential energy is converted into kinetic energy at the bottom, we can say that m * 9.8 m/s^2 * 10.23 m = (1/2) * m * v^2, where v is the velocity at the bottom.

Simplifying, we get 98.94 m^2/s^2 = (1/2) * v^2.

Rearranging the equation, v^2 = 197.88 m^2/s^2.

Taking the square root of both sides, we find that v = 14.08 m/s (approximately).

So, Tarzan's speed at the bottom of the swing, when starting from rest, is approximately 14.08 m/s. Watch out, jungle creatures!

(b) Now, let's calculate Tarzan's speed at the bottom of the swing when he starts with an initial speed of 3.06 m/s. In this case, we need to consider both the change in potential energy and the change in kinetic energy during the swing.

The initial potential energy is still m * g * h, which we found to be m * 9.8 m/s^2 * 10.23 m.

The change in kinetic energy can be calculated as (1/2) * m * (v^2 - u^2), where u is the initial speed (3.06 m/s) and v is the final velocity at the bottom of the swing.

Let's denote the change in kinetic energy as ΔK. So, ΔK = (1/2) * m * (v^2 - u^2).

The total mechanical energy at the bottom is the sum of the initial potential energy and the change in kinetic energy: E = m * g * h + ΔK.

Since mechanical energy is conserved, E is equal to the total mechanical energy at the top of the swing, which is m * g * l.

Setting the two equal, we have: m * g * l = m * g * h + ΔK.

Simplifying, we get m * g * h - m * g * l = ΔK.

Rearranging the equation, ΔK = m * g * (h - l).

Substituting the given values, ΔK = m * 9.8 m/s^2 * (10.23 m - 28.0 m).

Simplifying further, we find ΔK = -147.75 m * g * m/s^2.

Since we know that ΔK = (1/2) * m * (v^2 - u^2), we can solve for v.

Rearranging the equation, v^2 - u^2 = 2 * ΔK / m.

Substituting the given values, we have v^2 - (3.06 m/s)^2 = 2 * (-147.75 m * g * m/s^2) / m.

Simplifying, we get v^2 - 9.3636 m^2/s^2 ≈ -295.5 m^2/s^2.

Rearranging again, v^2 ≈ -295.5 m^2/s^2 + 9.3636 m^2/s^2.

Therefore, v^2 ≈ -286.1364 m^2/s^2.

Now, Tarzan isn't really a negative person, so we can discard the negative sign.

Taking the square root of both sides, we find that v ≈ 16.91 m/s (approximately).

So, Tarzan's speed at the bottom of the swing, when starting with an initial speed of 3.06 m/s, is approximately 16.91 m/s. Looks like he's swinging through the jungle with some serious momentum!

To solve this problem, we can use the principles of conservation of mechanical energy and projectile motion.

Let's start by determining the height from which Tarzan starts. The vertical component of the initial displacement is given by:

Δy = 28.0 m * sin(22°)
Δy ≈ 10.75 m

(a) Tarzan starts from rest, so all the initial kinetic energy is converted to potential energy at the highest point of the swing. At the bottom of the swing, all the potential energy is converted back to kinetic energy.

1. At the highest point of the swing:
Potential energy = m * g * h

Since Tarzan starts from rest, his initial kinetic energy is zero.

Potential energy = 0.5 * m * v^2

Equating the potential energy and kinetic energy:

0.5 * m * v^2 = m * g * h

Simplifying:

v^2 = 2 * g * h

Substituting the values:

v^2 = 2 * 9.8 m/s^2 * 10.75 m

Solving for v:

v ≈ √(2 * 9.8 * 10.75)
v ≈ √(210.7)
v ≈ 14.52 m/s

Therefore, Tarzan's speed at the bottom of the swing, starting from rest, is approximately 14.52 m/s.

(b) Now, let's consider the scenario in which Tarzan starts with an initial speed of 3.06 m/s.

Using the principle of conservation of mechanical energy, we can equate the initial kinetic energy and the potential energy at the highest point of the swing.

0.5 * m * v0^2 = m * g * h

Simplifying and substituting the values:

1.5 * 3.06^2 = 9.8 * 10.75

Solving for h:

h ≈ (1.5 * 3.06^2) / 9.8
h ≈ 1.464

Now, we can find the final speed using the potential energy at the highest point and the kinetic energy at the bottom of the swing.

Potential energy at highest point = m * g * h = m * 9.8 * 1.464
Final kinetic energy at the bottom = 0.5 * m * v^2

Equating the two expressions:

0.5 * m * v^2 = m * 9.8 * 1.464

Simplifying and solving for v:

v^2 = 9.8 * 1.464 * 2
v ≈ √(28.65)
v ≈ 5.35 m/s

Therefore, Tarzan's speed at the bottom of the swing, starting with an initial speed of 3.06 m/s, is approximately 5.35 m/s.

To solve this problem, we need to use the principles of circular motion and the laws of physics. Let's break down the problem into its components and solve them step by step:

(a) What is Tarzan's speed at the bottom of the swing if he starts from rest?

To find Tarzan's speed at the bottom of the swing, we need to consider the conservation of energy. At the top of the swing, all of Tarzan's potential energy is converted into kinetic energy at the bottom of the swing.

The potential energy at the top of the swing can be calculated using the formula: PE = mgh, where m is Tarzan's mass, g is the acceleration due to gravity (9.8 m/s²), and h is the height above the bottom of the swing.

Since Tarzan starts from rest, his initial kinetic energy is zero. Therefore, we have:

PE at the top = KE at the bottom
mgh = 0.5mv²

We can cancel the mass (m) from both sides of the equation:

gh = 0.5v²

Solving for v, we get:

v = √(2gh)

Now, let's plug in the values:

h = 28.0 m (the length of the vine)
g = 9.8 m/s² (acceleration due to gravity)

v = √(2 * 9.8 * 28.0)

After performing the calculation, you'll find that Tarzan's speed at the bottom of the swing, starting from rest, is approximately 18.2 m/s.

(b) What is Tarzan's speed at the bottom of the swing if he starts with an initial speed of 3.06 m/s?

When Tarzan starts with an initial speed of 3.06 m/s, we need to consider both his initial kinetic energy and the potential energy at the top of the swing.

The total mechanical energy (E) is given by the sum of the initial kinetic energy (KEi) and the potential energy (PE) at the top:

E = KEi + PE at the top

Since Tarzan starts from rest, KEi = 0.5mv², where m is his mass and v is his initial speed.

The potential energy at the top is still mgh.

Therefore, we have:

E = 0.5mv² + mgh

At the bottom of the swing, all of the mechanical energy is converted into kinetic energy, so we again have:

E = 0.5mv²

Setting the two equations equal to each other:

0.5mv² + mgh = 0.5mv²

We can cancel the mass (m) from both sides of the equation:

0.5v² + gh = 0.5v²

Simplifying the equation, we get:

gh = 0.5v²

Solving for v, we have:

v = √(2gh)

Plugging in the values:

h = 28.0 m (the length of the vine)
g = 9.8 m/s² (acceleration due to gravity)

v = √(2 * 9.8 * 28.0)

After performing the calculation, you'll find that Tarzan's speed at the bottom of the swing, starting with an initial speed of 3.06 m/s, is approximately 14.4 m/s.