Tarzan swings on a 27.0 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?
(b) What is his speed at the bottom of the swing if he pushes off with a speed of 3.00 m/s?
I tried to use energy equations and got the answer of final velocity = the square root of 2gh
But I don't think I am using the right thing for h.
please help
(a) If you use V = sqrt (2 g h), which is the correct formula, the h should be
27m *(1 - cos33)
(b) Add M g h to the initial kinetic energy, and solve for the new V. The mass M cancels out.
V^2/2 = Vo^2/2 + g*h
Vo is the "pushoff speed", 3.00 m/s.
Thank you SO much! I didn't realize i had to do 27(1-cos33)
To solve this problem, we can use the conservation of mechanical energy. The initial mechanical energy is equal to the final mechanical energy. Let's break it down step-by-step:
(a) When Tarzan starts from rest, the initial kinetic energy is zero. Therefore, all the initial mechanical energy is in the form of potential energy:
Initial mechanical energy = mgh
Here, m is the mass of Tarzan and g is the acceleration due to gravity.
To find the height h, we can use trigonometry:
h = L * sin(θ)
Where L is the length of the vine and θ is the angle it makes with the vertical.
So the initial mechanical energy is:
Initial mechanical energy = m * g * L * sin(θ)
Now, at the bottom of the swing, all the initial potential energy is converted to kinetic energy.
Final kinetic energy = (1/2) * m * v^2
Where v is the final velocity at the bottom of the swing.
Setting the initial mechanical energy equal to the final kinetic energy:
m * g * L * sin(θ) = (1/2) * m * v^2
Simplifying and solving for v:
v^2 = 2 * g * L * sin(θ)
Taking the square root of both sides:
v = √(2 * g * L * sin(θ))
Plugging in the given values:
g = 9.8 m/s^2
L = 27.0 m
θ = 33.0°
v = √(2 * 9.8 * 27.0 * sin(33.0))
Now you can calculate the value of v.
(b) If Tarzan pushes off with a speed of 3.00 m/s, then he already has some initial kinetic energy. The initial mechanical energy is the sum of the initial kinetic energy and potential energy:
Initial mechanical energy = (1/2) * m * (3.00)^2 + m * g * L * sin(θ)
At the bottom of the swing, the total mechanical energy remains the same:
(1/2) * m * (3.00)^2 + m * g * L * sin(θ) = (1/2) * m * v^2
Simplifying and solving for v:
v^2 = (3.00)^2 + 2 * g * L * sin(θ)
v = √[(3.00)^2 + 2 * 9.8 * 27.0 * sin(33.0)]
Now you can calculate the value of v for this case as well.
To solve this problem using energy equations, you are on the right track. The equation you mentioned, final velocity = √(2gh), can be used to find the final velocity of an object in free fall. However, the value of h you mentioned is incorrect in this case.
Let's go through the steps to solve this problem correctly:
(a) When Tarzan starts from rest, the only form of energy he has is potential energy. As he swings down, the potential energy will be converted into kinetic energy at the bottom of the swing. We can equate the potential energy to the kinetic energy using the conservation of energy.
The potential energy of Tarzan at the top of the swing can be calculated using the height, h, above the lowest point of the swing. In this case, h is equal to the length of the vine, 27.0 m, multiplied by the sin(33.0°) because the vine is inclined at an angle of 33.0° with the vertical. Thus, h = 27.0 m * sin(33.0°).
Therefore, the potential energy at the top of the swing is (mgh), where m is Tarzan's mass (not given) and g is the acceleration due to gravity (9.8 m/s²).
At the bottom of the swing, all the potential energy is converted into kinetic energy. So, we have (1/2)mv² = mgh.
Given that Tarzan starts from rest, his initial velocity, vi, is 0 m/s.
Using the equation (1/2)mv² = mgh, we can rewrite it as (1/2)mv² = mgh - mghₜ, where hₜ is the height at the top of the swing.
Since Tarzan starts from rest, he has no initial kinetic energy, so (1/2)mvᵢ² is zero.
Therefore, we have 0 = mgh - mghₜ.
Dividing both sides of the equation by m, we get 0 = gh - ghₜ.
Simplifying the equation, we have gh = ghₜ.
Now, substitute in the values to solve for the final velocity, v₂:
gh = ghₜ
9.8 m/s² * h = 9.8 m/s² * hₜ
h = 27.0 m * sin(33.0°)
h = 27.0 m * 0.549
h = 14.823 m
Simplifying the equation further:
9.8 m/s² * h = 9.8 m/s² * hₜ
9.8 m/s² * 14.823 m = 9.8 m/s² * hₜ
144.8556 = 9.8 * hₜ
Now, solve for hₜ:
hₜ = 144.8556 / 9.8
hₜ ≈ 14.782 m
Now we can substitute the values back into the energy equation to find the final velocity:
(1/2)mv₂² = mgh - mghₜ
(1/2)v₂² = gh - ghₜ
v₂² = 2gh - 2ghₜ
v₂ = √(2gh - 2ghₜ)
Substituting the known values, we get:
v₂ = √(2 * 9.8 m/s² * 14.823 m - 2 * 9.8 m/s² * 14.782 m)
v₂ ≈ √(291.0456 - 289.8124)
v₂ ≈ √1.2332
v₂ ≈ 1.11 m/s
So, Tarzan's speed at the bottom of the swing, starting from rest, is approximately 1.11 m/s.
(b) Now let's solve for the speed at the bottom of the swing if he pushes off with a speed of 3.00 m/s.
In this case, Tarzan starts with an initial kinetic energy of (1/2)mvᵢ² = (1/2)m(3.00 m/s)². Remember that there is no initial potential energy because he is starting at the bottom of the swing.
Using the same energy equation, (1/2)mv₂² = mgh - mghₜ, we substitute in the known values:
(1/2)m(3.00 m/s)² = 9.8 m/s² * h - 9.8 m/s² * 14.782 m
Simplifying the equation:
(1/2)(3.00 m/s)² = 9.8 m/s² * (h - 14.782 m)
(1/2)(9.00 m²/s²) = 9.8 m/s² * (h - 14.782 m)
4.50 = 9.8 m/s² * (h - 14.782 m)
Now, solving for h:
4.50 = 9.8 m/s² * (h - 14.782 m)
h - 14.782 m = 4.50 / 9.8
h ≈ 4.50 / 9.8 + 14.782 m
h ≈ 0.459 m + 14.782 m
h ≈ 15.241 m
Substituting the known values into the energy equation to find the final velocity:
v₂ = √(2gh - 2ghₜ)
v₂ = √(2 * 9.8 m/s² * 27.0 m * sin(33.0°) - 2 * 9.8 m/s² * 15.241 m)
v₂ ≈ √(256.092 - 299.7694)
v₂ ≈ √(-43.6774)
Since the square root of a negative number is not possible, there is no real answer in this case.