Tarzan is running at 7.5 m/s when he grabs a hanging vine from a tree branch. How high can he swing upward?
kinetic energy = (1/2) m (7.5)^2
Potential Energy distance h above running height = m g h = m (9.81) h
so
9.81 h = .5 * 7.5 * 7.5
solve for h
To determine how high Tarzan can swing upward, we need to consider the conservation of mechanical energy.
The mechanical energy of Tarzan consists of two parts: his kinetic energy (due to his motion) and his potential energy (due to his height).
1. First, we need to determine Tarzan's kinetic energy. The formula for kinetic energy is:
KE = (1/2) * m * v^2
where KE is the kinetic energy, m is the mass of Tarzan (which we'll assume is constant), and v is Tarzan's velocity.
Let's assume that Tarzan's mass is 70 kg, and his velocity is 7.5 m/s. Plugging these values into the formula, we get:
KE = (1/2) * 70 * (7.5)^2
= 1968.75 J
2. Since Tarzan grabs the vine, the total mechanical energy (sum of kinetic and potential energy) remains the same. As he swings upward, his kinetic energy will decrease, and his potential energy will increase.
At the highest point of the swing, Tarzan's velocity will be zero. Therefore, his kinetic energy is zero.
So, the total mechanical energy at the highest point is equal to Tarzan's potential energy:
PE = m * g * h
where PE is the potential energy, m is Tarzan's mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height Tarzan swings upward.
Rearranging the formula, we can solve for h:
h = PE / (m * g)
Plugging in the values, we get:
h = 1968.75 J / (70 kg * 9.8 m/s^2)
≈ 2.8 m
Therefore, Tarzan can swing upward to a height of approximately 2.8 meters.
To determine how high Tarzan can swing upward, we need to consider the principle of conservation of energy. The initial energy Tarzan has while running is converted into potential energy when he swings upward on the vine.
The formula for potential energy (PE) is:
PE = m * g * h
where m is the mass of Tarzan, g is the acceleration due to gravity (approximately 9.8 m/s² on Earth), and h is the height.
In this case, we don't have Tarzan's mass provided, but fortunately, we can assume that Tarzan's mass cancels out when comparing two situations involving the same person (assuming Tarzan has a consistent mass throughout the scenario).
Since we don't have the mass, we can rewrite the formula as:
PE/m = g * h
Now, let's solve for h:
h = PE / (m * g)
To calculate the potential energy, we need Tarzan's velocity at the highest point of the swing. At this point, all of his kinetic energy will be converted into potential energy. When Tarzan is at the highest point, his velocity will be zero.
The formula for kinetic energy (KE) is:
KE = 0.5 * m * v²
where v is Tarzan's velocity.
Since Tarzan's velocity is 7.5 m/s before grabbing the vine, we can calculate the kinetic energy:
KE = 0.5 * m * (7.5 m/s)²
Now we can use this value to find the potential energy at the highest point:
PE = KE
Substituting the values, we get:
PE = 0.5 * m * (7.5 m/s)²
Finally, we can substitute the potential energy value back into the formula for height:
h = PE / (m * g)
h = 0.5 * m * (7.5 m/s)² / (m * g)
Now we can simplify the equation to find the maximum height Tarzan can swing upward.