Tarzan (m = 78.4 kg) tries to cross a river by swinging from a vine. The vine is 10 m
long. Tarzan doesn't know that the vine has a breaking strength of 1045 N. What
maximum speed (in m/s) can Tarzan have at the bottom of the swing (as he just clears
the water) in order to safely cross the river without breaking the vine?
PLEASE LEAVE FULL ANSWERS!
At the bottom of the swing, the vine is subject to two forces,
gravity force = mg
centripetal force = mv^sup2;/r
m=mass of Tarzan (kg)
g=acceleration due to gravity (9.81 m/s²)
v=tangential velocity (m/s)
r=radius of circular motion
=length of vine (m)
At point of rupture of the vine,
mg+mv²/r = 1045 N
Solve for v.
Well, Tarzan really needs to "branch" out his knowledge on physics to swing safely across the river. Let's calculate the maximum speed he can have without losing his grip on the vine.
The potential energy at the highest point of the swing (just before he starts descending) will be converted into kinetic energy at the lowest point (as he clears the water).
The potential energy can be calculated using the formula: PE = m * g * h, where m is Tarzan's mass (78.4 kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of Tarzan at the top of the swing.
Since Tarzan starts at the highest point of the swing, all the potential energy will be converted into kinetic energy at the bottom of the swing. Therefore, we can say that the maximum speed he can have is equal to the square root of two times the gravitational acceleration times the swing height.
We can write this as an equation:
(1/2) * m * v_max^2 = m * g * h
Simplifying the equation:
(1/2) * v_max^2 = g * h
Now, we can solve for the maximum speed, v_max:
v_max = sqrt(2 * g * h)
Since the vine has a breaking strength of 1045 N, we can equate this force to the tension in the vine, which can be calculated as the mass of Tarzan times his acceleration at the bottom of the swing.
The tension in the vine (T) can be calculated using the formula: T = m * a
Given that the length of the vine is 10 m, Tarzan's maximum speed is v_max, and the tension in the vine is equal to its breaking strength (1045 N), we can find the acceleration (a) at the bottom of the swing using the equation: a = v_max^2 / L
Substituting the values, we have: 1045 N = 78.4 kg * (v_max^2 / 10 m)
Now, let's solve for v_max:
v_max^2 = 1045 N * 10 m / 78.4 kg
v_max^2 = 133.335 m²/s²
Finally, taking the square root of both sides, we get:
v_max = sqrt(133.335 m²/s²) = 11.55 m/s
So, Tarzan can safely cross the river without breaking the vine as long as his maximum speed at the bottom of the swing is approximately 11.55 m/s. Just remember, Tarzan, don't "vine" about going too fast!
In order to solve this problem, we need to consider the forces acting on Tarzan as he swings on the vine.
1. Weight Force: Tarzan's weight can be calculated using the formula:
Fg = m * g
where Fg is the weight force, m is Tarzan's mass, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the values, we get:
Fg = 78.4 kg * 9.8 m/s^2
= 768.32 N
2. Centripetal Force: As Tarzan swings on the vine, there is a centripetal force acting towards the center of the swing. This force can be calculated using the formula:
Fc = (m * v^2) / r
where Fc is the centripetal force, m is Tarzan's mass, v is his speed, and r is the radius of the swing (half the length of the vine).
Plugging in the values, we get:
Fc = (78.4 kg * v^2) / (10 m / 2)
= (78.4 kg * v^2) / 5 m
= 15.68 kg * v^2
Tarzan can safely swing on the vine if the centripetal force is less than or equal to the breaking strength of the vine, which is given as 1045 N. Therefore, we can set up the following inequality:
Fc ≤ Breaking Strength
15.68 kg * v^2 ≤ 1045 N
Now we can solve this inequality to find the maximum speed (v) at which Tarzan can swing.
v^2 ≤ 66.5 N / kg
v ≤ √(66.5 N / kg)
Calculating the square root gives:
v ≤ √(66.5 N / kg)
≈ 8.16 m/s
Therefore, the maximum speed Tarzan can have at the bottom of the swing in order to safely cross the river without breaking the vine is approximately 8.16 m/s.
To determine the maximum speed at the bottom of the swing that Tarzan can have without breaking the vine, we need to consider the forces acting on him.
First, let's analyze the forces during the swing. At the bottom of the swing, Tarzan's weight is perpendicular to the vine, meaning it does not contribute to the tension in the vine. The only force acting on Tarzan is the tension in the vine.
The tension in the vine can be calculated using the centripetal force formula:
Tension = Mass × Centripetal acceleration.
Since Tarzan is swinging in a circular path, his acceleration is the centripetal acceleration, given by:
Centripetal acceleration = (velocity^2) / radius.
The radius in this case is the length of the vine, which is 10 m.
So, the tension in the vine can be expressed as:
Tension = Mass × [(velocity^2) / radius].
We know that the tension must not exceed the breaking strength of the vine, which is 1045 N. Therefore, we can set up the following equation:
Mass × [(velocity^2) / radius] ≤ Breaking strength of the vine.
Plugging in the given values:
78.4 kg × [(velocity^2) / 10 m] ≤ 1045 N.
Simplifying the equation, we have:
7.84 kg × velocity^2 ≤ 1045 N.
Dividing both sides by 7.84 kg, we get:
velocity^2 ≤ 133.2 m^2/s^2.
Taking the square root of both sides, we find:
velocity ≤ √133.2 ≈ 11.54 m/s.
Therefore, the maximum speed at the bottom of the swing that Tarzan can have without breaking the vine is approximately 11.54 m/s.