Tarzan swings on a 33.0 m long vine initially
inclined at an angle of 31.0◦ with the vertical.
The acceleration of gravity if 9.81 m/s
What is his speed at the bottom of the
swing if he
a) starts from rest?
Answer in units of m/s
b) pushes off with a speed of 5.00 m/s?
Answer in units of m/s.
Thanks!
Why you put 12(1-cos 31degrees)
a) If Tarzan starts from rest, he will gain potential energy as he swings down and convert it into kinetic energy at the bottom of the swing. We can use the conservation of energy to solve for his speed.
At the top of the swing, all the energy is potential energy (PE) and at the bottom of the swing, all the energy is kinetic energy (KE).
PE = mgh
KE = 0.5mv^2
At the top of the swing, the potential energy is given by PE = mgh, where m is the mass of Tarzan (which we can assume to be canceled out in this case), g is the acceleration due to gravity, and h is the height of the swing (which is equal to the length of the vine).
PE = mgh = mgh
At the bottom of the swing, the potential energy is zero and the entire energy is converted into kinetic energy.
KE = 0.5mv^2
Setting the potential energy at the top equal to the kinetic energy at the bottom, we get:
mgh = 0.5mv^2
Canceling out the mass, we have:
gh = 0.5v^2
Solving for v, we get:
v = √(2gh)
Plugging in the values, we have:
v = √(2 * 9.81 * 33)
v ≈ √(649.26)
v ≈ 25.48 m/s
Therefore, his speed at the bottom of the swing if he starts from rest is approximately 25.48 m/s.
b) If Tarzan pushes off with a speed of 5.00 m/s, his initial kinetic energy would contribute to his total energy at the bottom of the swing.
To find his speed at the bottom, we need to add his initial kinetic energy to the kinetic energy at the bottom of the swing.
KE_initial = 0.5mv_initial^2
KE_bottom = 0.5mv_bottom^2
The total energy at the bottom is the sum of these two energies:
Total Energy at the bottom = KE_initial + KE_bottom
Total Energy at the bottom = 0.5mv_initial^2 + 0.5mv_bottom^2
At the bottom, the potential energy is zero, so the total energy is equal to the kinetic energy:
Total Energy at the bottom = KE_bottom
Setting the expression for the total energy equal to the expression for the kinetic energy at the bottom, we have:
0.5mv_initial^2 + 0.5mv_bottom^2 = 0.5mv_bottom^2
Canceling out the mass, we get:
0.5v_initial^2 + 0.5v_bottom^2 = 0.5v_bottom^2
Rearranging the terms, we have:
0.5v_initial^2 = 0
Simplifying further, we have:
v_initial^2 = 0
Taking the square root of both sides, we have:
v_initial = 0
Therefore, if Tarzan pushes off with a speed of 5.00 m/s, his speed at the bottom remains the same, which is approximately 5.00 m/s.
To find Tarzan's speed at the bottom of the swing, we can use the principles of conservation of mechanical energy. At the bottom of the swing, Tarzan's potential energy is converted to kinetic energy.
a) If Tarzan starts from rest, his initial kinetic energy is zero. Therefore, his initial potential energy is equal to his total mechanical energy.
To find the potential energy, we can use the formula:
Potential Energy = m * g * h
where m is Tarzan's mass, g is the acceleration due to gravity, and h is the height of Tarzan at the starting point.
Since Tarzan is starting from rest, his initial kinetic energy is zero. Therefore, his initial potential energy is converted to his final kinetic energy at the bottom of the swing.
At the bottom of the swing, Tarzan's potential energy is zero, and his kinetic energy is at a maximum. Therefore, we can equate his initial potential energy to his final kinetic energy:
m * g * h = (1/2) * m * v^2
where v is the final velocity at the bottom of the swing.
We can cancel out the mass (m) from both sides of the equation, yielding:
g * h = (1/2) * v^2
To find v, we can rearrange the equation:
v = √(2 * g * h)
Now we can substitute the given values:
g = 9.81 m/s^2
h = 33.0 m
Plugging these values into the equation, we get:
v = √(2 * 9.81 * 33.0) ≈ 28.94 m/s
Therefore, Tarzan's speed at the bottom of the swing, when he starts from rest, is approximately 28.94 m/s.
b) If Tarzan pushes off with a speed of 5.00 m/s, we need to add this initial kinetic energy to the final kinetic energy at the bottom of the swing.
Using the same conservation of mechanical energy equation, we get:
(m * g * h) + (1/2) * m * (5.00)^2 = (1/2) * m * v^2
Cancelling out the mass (m), we have:
g * h + (1/2) * (5.00)^2 = (1/2) * v^2
To solve for v, we can rearrange the equation:
v = √(2 * (g * h + (1/2) * (5.00)^2))
Plugging in the values:
g = 9.81 m/s^2
h = 33.0 m
We get:
v = √(2 * (9.81 * 33.0 + (1/2) * (5.00)^2)) ≈ 29.03 m/s
Therefore, Tarzan's speed at the bottom of the swing, when he starts with an initial speed of 5.00 m/s, is approximately 29.03 m/s.
height above bottom call it h
h = 33(1-cos 31)
initial energy = m g h
speed at bottom = sqrt(2gh)
because
(1/2)mv^2 = m g h
for part b
initial energy = m g h +(1/2)m(25)
so at bottom
(1/2) m v^2 = that initial energy