Tarzan swings on a vine from a cliff to rescue Jane, who is standing on the ground surrounded by snakes. His plan is to push off the cliff, grab Jane at the lowest point of his swing, and carry them both to the safety of a nearby tree (see figure). Tarzan's mass is 76.1 kg, Jane's mass is 43.3 kg, the height of the lowest limb of the target tree is 11.5 m, and Tarzan is initially standing on a cliff of height 20.9 m. The length of the vine is 34.5 m. With what speed should Tarzan push off the cliff if he and Jane are to make it to the tree limb successfully?
We can start by first finding how high up Tarzan and Jane will be relative to the lowest limb of the target tree when they reach it, since they will need to have this height in order for Tarzan to grab the limb and ensure their safety.
The tree limb is 11.5 m high, and Tarzan starts at a height of 20.9 m. The length of the vine is 34.5 m, so when Tarzan is at the lowest point of his swing, he will be at the height of the difference between the vine length and the cliff height, $34.5 - 20.9 = 13.6\text{ m}$. Since the target tree limb is at 11.5 m, Tarzan and Jane need to be 13.6 - 11.5 = 2.1 m above the limb to safely grab it.
Now let's consider the conservation of mechanical energy. When Tarzan pushes off the cliff, he converts his potential energy (mass x gravity x height) into kinetic energy (1/2 x mass x velocity^2). Since he picks up Jane, we need to include her mass in the total mass for the calculations once Tarzan grabs her.
At the beginning, Tarzan's initial potential energy (PE) can be calculated using: $PE = mgh$, where $m = 76.1\text{ kg}$, $g = 9.81\text{ m/s}^2$, and $h = 20.9\text{ m}$. Therefore, $PE = 76.1 \times 9.81 \times 20.9 = 15,\!850.9\text{ J}$ (joules).
Since Tarzan will pick up Jane at the lowest point of his swing, their total mass will be $76.1 + 43.3 = 119.4\text{ kg}$. When they reach the 2.1 m above the limb, their total potential energy (PE) can be calculated using: $PE = mgh$, where $m = 119.4\text{ kg}$, $g = 9.81\text{ m/s}^2$, and $h = 11.5 + 2.1 = 13.6\text{ m}$. Therefore, $PE = 119.4 \times 9.81 \times 13.6 = 16,\!862.1\text{ J}$.
Since the mechanical energy is conserved, we can find the kinetic energy of Tarzan and Jane at the 2.1 m above the limb, which is the same as their potential energy at this point, $16,\!862.1\text{ J}$.
The kinetic energy equation is $KE = 1/2 \times m \times v^2$, where $v$ is the velocity Tarzan must push off the cliff with. For Tarzan and Jane, the total mass is $119.4\text{ kg}$ and $KE = 16,\!862.1\text{ J}$. This equation can be solved for the velocity:
$v^2 = 2 \times 16,\!862.1\text{ J} / 119.4\text{ kg} = 282.1\text{ (m^2/s^2)}$
$v = \sqrt{282.1} \approx 16.8\text{ m/s}$
So, Tarzan should push off the cliff with a speed of about $16.8\text{ m/s}$ to successfully carry Jane to the safety of the nearby tree limb.
To solve this problem, we can use the principle of conservation of mechanical energy.
1. First, let's calculate the gravitational potential energy of Tarzan and Jane at the starting point on the cliff.
Potential Energy (Tarzan) = m * g * h
Potential Energy (Tarzan) = 76.1 kg * 9.8 m/s^2 * 20.9 m
2. Next, let's calculate the gravitational potential energy of Tarzan and Jane at the lowest point of their swing.
Potential Energy (Tarzan) = m * g * h
Potential Energy (Tarzan) = 76.1 kg * 9.8 m/s^2 * 11.5 m
3. The difference in potential energy between the starting point and the lowest point is equal to the kinetic energy at the lowest point.
Potential Energy (Tarzan) - Potential Energy (Tarzan) = 1/2 * (m + m) * v^2
(76.1 kg * 9.8 m/s^2 * 20.9 m) - (76.1 kg * 9.8 m/s^2 * 11.5 m) = 1/2 * (76.1 kg + 43.3 kg) * v^2
4. Simplify the equation and solve for v (the velocity at the lowest point).
(76.1 kg * 9.8 m/s^2 * 20.9 m) - (76.1 kg * 9.8 m/s^2 * 11.5 m) = 1/2 * (76.1 kg + 43.3 kg) * v^2
5. Calculate v by dividing both sides of the equation by (1/2 * (76.1 kg + 43.3 kg)).
v^2 = [(76.1 kg * 9.8 m/s^2 * 20.9 m) - (76.1 kg * 9.8 m/s^2 * 11.5 m)] / (1/2 * (76.1 kg + 43.3 kg))
6. Take the square root of both sides to solve for v.
v = sqrt{[(76.1 kg * 9.8 m/s^2 * 20.9 m) - (76.1 kg * 9.8 m/s^2 * 11.5 m)] / (1/2 * (76.1 kg + 43.3 kg))}
7. Calculate the value of v using a calculator.
v ≈ sqrt{(148329.68 J - 85504.47 J) / (119.4 kg)}
v ≈ sqrt{(62825.21 J) / (119.4 kg)}
v ≈ sqrt{525.97 m^2/s^2} ≈ 22.95 m/s
Therefore, Tarzan should push off the cliff with a speed of approximately 22.95 m/s in order to successfully rescue Jane and reach the safety of the nearby tree limb.
To solve this problem, we can use the principles of conservation of mechanical energy.
First, let's consider the initial potential energy and the final potential energy of Tarzan and Jane.
The initial potential energy (PEi) of Tarzan is given by the product of his mass (m) and the height of the cliff (h):
PEi = m * h
Similarly, the initial potential energy (PEi) of Jane is given by:
PEi = m * h
The final potential energy (PEf) of Tarzan and Jane is given by the product of their total mass (m_t) and the height of the tree limb (h_t):
PEf = m_t * h_t
Since Tarzan and Jane are connected by the vine, the sum of their masses is used in the equation.
Now, let's consider the initial kinetic energy (KEi) and the final kinetic energy (KEf) of Tarzan and Jane.
The initial kinetic energy (KEi) of Tarzan is zero because he is initially at rest.
The final kinetic energy (KEf) of Tarzan and Jane is given by the product of their total mass (m_t) and the final velocity (v_f):
KEf = (1/2) * m_t * v_f^2
where v_f is the final velocity.
According to the conservation of mechanical energy, the initial potential energy plus the initial kinetic energy should be equal to the final potential energy plus the final kinetic energy:
PEi + KEi = PEf + KEf
Since Tarzan pushes off the cliff, his initial potential energy is converted into kinetic energy:
m * h = KEf
Simplifying this equation, we get:
m * g * h = KEf
where g is the acceleration due to gravity (9.8 m/s^2).
Now, let's substitute the values given in the problem:
m = 76.1 kg
h = 20.9 m
m_t = 76.1 kg + 43.3 kg = 119.4 kg
h_t = 11.5 m
g = 9.8 m/s^2
Substituting these values into the equation, we have:
76.1 kg * 9.8 m/s^2 * 20.9 m = (1/2) * 119.4 kg * v_f^2
Simplifying this equation, we can determine the final velocity (v_f) of Tarzan and Jane.