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?
draw the sketch. How high is he initially?
intialPE=finalKE
b) initialPE+ intialKE= finalKE
I drew the sketch, the problem does not tell me how high he is initially, I can find the equation vfinal = the square root of (2gh) but I don't know what h is.
I know this is about 5 years too late, but h would be the length of the vine (21 m)
Tarzan swings on a 27.0 m long vine initially inclined at an angle of 24° from the vertical.
What is his speed at the bottom of the swing if he starts from rest?
To find Tarzan's speed at the bottom of the swing, we can use the principles of conservation of energy and trigonometry.
(a) When Tarzan starts from rest (v = 0), the total mechanical energy at the bottom of the swing is equal to the initial potential energy. Assuming no energy loss due to friction or air resistance, we have:
Initial potential energy (PE) = Total mechanical energy (E)
The initial potential energy depends on Tarzan's mass (m), the acceleration due to gravity (g), and the height of Tarzan at the starting point. Since Tarzan starts swinging from the vertical position, the vertical height can be calculated as follows:
Height (h) = Length of the vine * sin(angle)
Substituting the given values:
h = 27.0 m * sin(33.0°)
Now, we can calculate the initial potential energy:
PE = m * g * h
Next, we equate the initial potential energy to the total mechanical energy at the bottom of the swing:
PE = E
Since the total mechanical energy includes both potential energy and kinetic energy, we can express it as:
E = PE + KE
At the bottom of the swing, when Tarzan is at the lowest point, all the potential energy is converted into kinetic energy. Therefore, we can rewrite the equation as:
PE = KE
Finally, we can calculate Tarzan's speed at the bottom of the swing using the principle of conservation of energy:
v = √(2 * g * h)
Substituting the values:
v = √(2 * 9.8 m/s^2 * (27.0 m * sin(33.0°)))
(b) If Tarzan pushes off with a speed of 3.00 m/s, we need to consider both the initial kinetic energy and the potential energy as he starts swinging. The total mechanical energy at the bottom of the swing is still given by:
Total mechanical energy (E) = Initial potential energy (PE) + Initial kinetic energy (KE)
Since Tarzan pushes off with a velocity of 3.00 m/s, his initial kinetic energy can be calculated as:
KE = (1/2) * m * v^2
Then, we equate the initial potential energy to the total mechanical energy at the bottom of the swing:
PE = E
And we have:
PE = KE
Solving for v:
v = √(2 * (PE - KE) / m)
Substituting the given values:
v = √(2 * 9.8 m/s^2 * (27.0 m * sin(33.0°) - (1/2) * m * (3.00 m/s)^2) / m)
Simplifying the equation, we can cancel out the mass term:
v = √(2 * 9.8 m/s^2 * (27.0 m * sin(33.0°) - (1/2) * (3.00 m/s)^2))