Tarzan swings on a 23.2 m long vine initially inclined at an angle of 22° from the vertical.
(a) What is his speed at the bottom of the swing if he starts from rest?
m/s
(b) What is his speed at the bottom of the swing if he starts with an initial speed of 2.58 m/s?
m/s
To find Tarzan's speed at the bottom of the swing, we need to use the concept of energy conservation. At the top of the swing, Tarzan has only potential energy, and at the bottom of the swing, he has both potential and kinetic energy.
(a) If Tarzan starts from rest, he only has potential energy at the top of the swing, which is given by the equation:
Potential energy = Mass * Gravity * Height
The mass of Tarzan cancels out, so the equation simplifies to:
Potential energy = Gravity * Height
Since the height is given as 23.2 m, and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the potential energy at the top.
Potential energy = 9.8 m/s^2 * 23.2 m = 214.56 J
At the bottom of the swing, Tarzan has both potential and kinetic energy. The total mechanical energy remains constant, so the potential energy at the top must be equal to the sum of potential and kinetic energy at the bottom.
At the bottom of the swing, the potential energy is zero, and the kinetic energy is given by:
Kinetic energy = 1/2 * Mass * Velocity^2
Since Tarzan's mass is not provided, we can solve for his velocity using the principle of conservation of energy.
Potential energy at the top = Kinetic energy at the bottom
Gravity * Height = 1/2 * Mass * Velocity^2
Rearranging the equation to solve for velocity:
Velocity = sqrt((2 * Gravity * Height) / Mass)
Calculating the velocity:
Velocity = sqrt((2 * 9.8 m/s^2 * 23.2 m) / Mass)
As the mass is not given, we cannot find the exact velocity. However, we can calculate the velocity in terms of mass by dividing both sides of the equation by the mass:
Velocity / sqrt(mass) = sqrt((2 * 9.8 m/s^2 * 23.2 m) / mass)
The velocity will vary depending on the mass of Tarzan, but you can plug in the value of mass to calculate the exact velocity.
(b) If Tarzan starts with an initial speed of 2.58 m/s, we can use the principle of conservation of energy to find his speed at the bottom of the swing.
The total mechanical energy remains constant, so the initial mechanical energy (potential and kinetic energy) must be equal to the mechanical energy at the bottom of the swing.
Initial mechanical energy = Potential energy at the top + Kinetic energy at the top
The potential energy at the top is the same as before:
Potential energy = Gravity * Height
Substituting the values:
Potential energy = 9.8 m/s^2 * 23.2 m = 214.56 J
The initial kinetic energy is given by:
Kinetic energy = 1/2 * Mass * Initial velocity^2
Substituting the given initial velocity of 2.58 m/s:
Kinetic energy = 1/2 * Mass * (2.58 m/s)^2 = 1.67 Mass
Since the total mechanical energy remains constant, this is equal to the sum of potential and kinetic energy at the bottom:
Total mechanical energy = Potential energy at the bottom + Kinetic energy at the bottom
At the bottom of the swing, the potential energy is zero, and the kinetic energy is given by:
Kinetic energy = 1/2 * Mass * Final velocity^2
Substituting the equation:
214.56 J + 1.67 Mass = 1/2 * Mass * Final velocity^2
Rearranging the equation:
Final velocity = sqrt((214.56 J) / (1/2 * Mass)) = sqrt((429.12 J) / Mass)
The final velocity depends on the mass of Tarzan, so you would need to know the exact mass to calculate the final velocity precisely.