Tarzan (80 kg) runs toward a vine at 4.8 m/s where Jane (60 kg) is hanging on. He jumps and grabs the vine. How high does the vine swing? If the vine is 25 m long, how much of an angle does that make?
To find out how high the vine swings, we need to consider the conservation of momentum. Initially, both Tarzan and Jane are stationary and have no momentum. When Tarzan jumps and grabs onto the vine, he transfers his momentum to the vine and Jane.
Let's break down the problem step by step:
Step 1: Find Tarzan's initial momentum
Tarzan's initial momentum can be calculated using the formula: momentum = mass × velocity
Tarzan's mass is 80 kg, and his velocity is 4.8 m/s.
Thus, his initial momentum is: momentum = 80 kg × 4.8 m/s = 384 kg⋅m/s
Step 2: Consider the conservation of momentum
Since Tarzan jumps and grabs the vine, we can apply the principle of conservation of momentum. The total momentum before the jump (when both Tarzan and Jane are stationary) must be equal to the total momentum after the jump (when Tarzan and Jane move together on the vine).
Since Jane was initially stationary, her momentum will be zero.
Hence, the total momentum after the jump is equal to Tarzan's initial momentum.
Step 3: Calculate the combined mass of Tarzan and Jane
To calculate the combined mass, we need to add Tarzan's and Jane's masses together.
Tarzan's mass is 80 kg, and Jane's mass is 60 kg.
Thus, the combined mass is: 80 kg + 60 kg = 140 kg.
Step 4: Calculate the velocity of Tarzan and Jane after the jump
Since momentum is conserved, the total momentum after the jump is equal to the initial momentum.
The total momentum (after the jump) is: 384 kg⋅m/s.
To find the velocity of both Tarzan and Jane together, divide the total momentum by the combined mass:
velocity = total momentum ÷ combined mass
velocity = 384 kg⋅m/s ÷ 140 kg ≈ 2.7429 m/s
Step 5: Calculate the potential energy of Tarzan and Jane
When Tarzan and Jane reach their highest point, all of their initial kinetic energy is converted into potential energy.
The potential energy can be calculated by using the equation: potential energy = mass × gravity × height
The mass is the combined mass of Tarzan and Jane, which is 140 kg.
The acceleration due to gravity is 9.8 m/s^2.
Let's assume the vine swings to a height "h."
Therefore, potential energy = 140 kg × 9.8 m/s^2 × h = 1372h J
Step 6: Calculate the height of the vine swing
Since all of the initial kinetic energy is converted into potential energy, we can equate the initial kinetic energy to the potential energy:
(1/2) × mass × velocity^2 = mass × gravity × height
Simplifying this equation:
(1/2) × 140 kg × (2.7429 m/s)^2 = 140 kg × 9.8 m/s^2 × h
(1/2) × 140 kg × 7.525 m^2/s^2 = 1372h J
Solving for h:
(1/2) × 140 kg × 7.525 m^2/s^2 = 1372h J
h = (1/2) × 140 kg × 7.525 m^2/s^2 ÷ 1372 J
h ≈ 2.4227 m
Therefore, the vine swings to a height of approximately 2.4227 meters.
To calculate the angle that the vine makes, we can use trigonometry since the vine forms a right triangle with the ground where the length of the vine is the hypotenuse and the height is the opposite side.
Step 7: Calculate the angle
Using the given information, the length of the vine is 25 meters, and the height is 2.4227 meters.
The tangent of an angle is defined as the ratio of the opposite side (height) to the adjacent side (length of the vine).
Therefore, the tangent of the angle can be calculated as: tan(angle) = opposite/adjacent = height/length of the vine = 2.4227/25
To find the angle, we can take the inverse tangent (arctan) of both sides:
angle = arctan(tan(angle)) = arctan(2.4227/25)
Using a calculator, the angle is approximately 5.584 degrees.
Therefore, the vine swings to a height of approximately 2.4227 meters and makes an angle of approximately 5.584 degrees with the ground.
To determine how high the vine swings, we need to calculate the potential energy gained by Tarzan and Jane as they swing on the vine.
First, let's calculate the potential energy gained by Tarzan using the formula:
Potential Energy = mass × gravity × height
Tarzan's mass is 80 kg, and the acceleration due to gravity is approximately 9.8 m/s². Let's assume Tarzan swings to a maximum height, and we'll solve for that.
Potential Energy gained by Tarzan = 80 kg × 9.8 m/s² × height
Next, let's calculate the potential energy gained by Jane. Following the same process as above:
Potential Energy gained by Jane = 60 kg × 9.8 m/s² × height
Since Tarzan and Jane are swinging together on the same vine, their potential energies will be equal. Therefore, we can equate the two potential energy equations:
80 kg × 9.8 m/s² × height = 60 kg × 9.8 m/s² × height
Simplifying the equation, we get:
80 kg × height = 60 kg × height
This implies that the height reached by Tarzan and Jane is the same.
Now that we have determined the height, let's calculate it using the given information about the vine's length.
We know the length of the vine is 25 m. When Tarzan jumps and grabs the vine, he will start from ground level. As he swings, the height he reaches on the vine will be the difference between the length of the vine and the initial height. Let's call the height gained on the vine "vine height."
Therefore:
vine height = vine length - initial height
As Tarzan starts from ground level, the initial height is 0. Hence:
vine height = 25 m - 0 m
vine height = 25 m
So, the height at which the vine swings is 25 meters.
To find the angle made by the vine, we can use the tangent function:
Tangent of the angle = vine height / vine length
Tangent of the angle = 25 m / 25 m
Tangent of the angle = 1
To find the angle, we use the inverse tangent (arctan) function:
Angle = arctan(1)
Angle = 45 degrees (approximately)
Therefore, the vine swing reaches a height of 25 meters, and the angle made by the vine is approximately 45 degrees.