Tarzan swings on a 34.0-m-long vine initially inclined at an angle of 33.0° with the vertical.

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

(b) What is his speed at the bottom of the swing if he pushes off with a speed of 5.00 m/s?
m/s

To find the speed of Tarzan at the bottom of the swing, we can use the principles of energy conservation. The potential energy at the top of the swing is converted to kinetic energy at the bottom of the swing.

(a) When Tarzan starts from rest, his initial kinetic energy is zero. Therefore, all the potential energy is converted to kinetic energy at the bottom of the swing.

The potential energy at the top of the swing can be determined using the equation:

Potential energy = mass x gravity x height

Assuming Tarzan has a mass of 70 kg and the acceleration due to gravity is approximately 9.8 m/s^2, the potential energy at the top of the swing is:

Potential energy = 70 kg x 9.8 m/s^2 x 34.0 m x sin(33.0°)

The kinetic energy at the bottom of the swing is equal to the potential energy at the top, since there is no initial kinetic energy. Therefore:

Kinetic energy = Potential energy

Using the equation for kinetic energy:

Kinetic energy = (1/2) x mass x velocity^2

Setting the potential energy equal to the kinetic energy and solving for velocity, we have:

(1/2) x mass x velocity^2 = 70 kg x 9.8 m/s^2 x 34.0 m x sin(33.0°)

Simplifying the equation, we can solve for velocity:

velocity^2 = (70 kg x 9.8 m/s^2 x 34.0 m x sin(33.0°)) / (0.5 x 70 kg)

velocity^2 = 9.8 m/s^2 x 34.0 m x sin(33.0°)

velocity^2 = 9.8 m/s^2 x 34.0 m x 0.544

velocity^2 = 166.1056

Taking the square root of both sides, we find:

velocity = √166.1056

velocity ≈ 12.88 m/s

Therefore, the speed of Tarzan at the bottom of the swing, starting from rest, is approximately 12.88 m/s.

(b) If Tarzan pushes off with an initial speed of 5.00 m/s, we need to consider the initial kinetic energy as well as the potential energy at the top of the swing.

The total mechanical energy is conserved throughout the swing, so we can add the initial kinetic energy to the potential energy at the top to find the kinetic energy at the bottom.

Using the same formula as before, the potential energy at the top of the swing is:

Potential energy = 70 kg x 9.8 m/s^2 x 34.0 m x sin(33.0°)

The initial kinetic energy is:

Initial kinetic energy = (1/2) x mass x initial velocity^2

Initial kinetic energy = (1/2) x 70 kg x (5.00 m/s)^2

The total kinetic energy at the bottom is then:

Total kinetic energy = Potential energy + Initial kinetic energy

Using the equation for kinetic energy, we can solve for velocity:

(1/2) x mass x velocity^2 = Potential energy + Initial kinetic energy

(1/2) x 70 kg x velocity^2 = 70 kg x 9.8 m/s^2 x 34.0 m x sin(33.0°) + (1/2) x 70 kg x (5.00 m/s)^2

Simplifying the equation and solving for velocity, we have:

velocity^2 = (70 kg x 9.8 m/s^2 x 34.0 m x sin(33.0°) + (1/2) x 70 kg x (5.00 m/s)^2) / (0.5 x 70 kg)

velocity^2 = (9.8 m/s^2 x 34.0 m x sin(33.0°) + (1/2) x (5.00 m/s)^2) / 0.5

velocity^2 = 221.8236

Taking the square root of both sides, we find:

velocity = √221.8236

velocity ≈ 14.89 m/s

Therefore, the speed of Tarzan at the bottom of the swing, when he pushes off with a speed of 5.00 m/s, is approximately 14.89 m/s.

To solve these problems, we need to consider the conservation of energy and the principle of conservation of mechanical energy.

(a) When Tarzan starts from rest, all of his initial energy is potential energy at the top of the swing, and at the bottom, all of his energy is kinetic energy.

We can calculate his speed at the bottom of the swing using the formula:

mgh = (1/2)mv^2

where m is the mass of Tarzan (which we assume to be negligible), g is the acceleration due to gravity (approximately 9.8 m/s^2), h is the height (which is equal to the length of the vine, L), and v is the speed.

Rearranging the equation, we get:

v^2 = 2gh

Plugging in the values, we have:

v^2 = 2 * 9.8 m/s^2 * 34.0 m * sin(33.0°)

To find the speed, we take the square root of both sides:

v = √(2 * 9.8 m/s^2 * 34.0 m * sin(33.0°))

By plugging in the values and performing the calculation, we can find the speed at the bottom of the swing.

(b) When Tarzan pushes off with a speed of 5.00 m/s, he already has some initial kinetic energy. In this case, we need to consider both the initial kinetic energy and the potential energy at the top of the swing.

The total mechanical energy is given by:

E = (1/2)mv^2 + mgh

where E is the total mechanical energy.

Since Tarzan starts from rest at the top of the swing, the potential energy at the top is equal to zero.

Therefore, the total mechanical energy is:

E = (1/2)m(5.00 m/s)^2 + 0

At the bottom of the swing, Tarzan's total mechanical energy is equal to his kinetic energy:

E = (1/2)m(v^2)

Equating these two expressions for the energy, we can solve for v:

(1/2)m(5.00 m/s)^2 + 0 = (1/2)m(v^2)

Simplifying the equation, we have:

(5.00 m/s)^2 = v^2

Taking the square root of both sides, we get:

5.00 m/s = v

Therefore, the speed at the bottom of the swing, given that Tarzan pushes off with a speed of 5.00 m/s, is 5.00 m/s.

(a) PE= mgh=mgL(1-cosα),

KE= mv²/2.
PE=KE
mgL(1-cosα)= mv²/2,
v=sqrt{2gL(1-cosα)}= ….
(b) PE+KE1=KE
mgL(1-cosα) + mv₀²/2= mv²/2,
v = sqrt{2gL(1-cosα) + v₀²}= …