Tarzan swings on a 30.0 - m - long vine initially inclined at an angle of 37.0 degrees with the vertical.
What is his speed at the bot- tom of the swing (a) if he starts from rest? (b) If he pushes off with a
speed of 4.00 m/s?
a) If Tarzan starts from rest, we can use conservation of mechanical energy to solve for his speed at the bottom of the swing. At the top of the swing, all of Tarzan's initial potential energy will be converted to kinetic energy at the bottom of the swing.
The potential energy at the top of the swing is given by:
PE = m * g * h
where m is Tarzan's mass, g is the acceleration due to gravity, and h is the vertical height Tarzan's center of mass reaches above his starting point.
Since the vine is inclined at an angle of 37.0 degrees with the vertical, we can find h using trigonometry:
h = 30.0 m * sin(37.0°)
The potential energy at the top is then:
PE = m * g * 30.0 m * sin(37.0°)
At the bottom of the swing, all of Tarzan's potential energy has been converted to kinetic energy:
KE = 1/2 * m * v^2
Setting the potential energy at the top equal to the kinetic energy at the bottom, we have:
m * g * 30.0 m * sin(37.0°) = 1/2 * m * v^2
Simplifying and solving for v, we get:
v = sqrt(2 * g * 30.0 m * sin(37.0°))
b) If Tarzan pushes off with a speed of 4.00 m/s, we can add the initial kinetic energy to the equation. The kinetic energy at the top of the swing is given by:
KE_initial = 1/2 * m * 4.00 m/s^2
Adding this to the equation, we have:
KE_initial + m * g * 30.0 m * sin(37.0°) = 1/2 * m * v^2
Simplifying and solving for v, we get:
v = sqrt(2 * (KE_initial + g * 30.0 m * sin(37.0°)))
So, the speed at the bottom of the swing depends on whether Tarzan starts from rest or pushes off with a speed of 4.00 m/s.
To find Tarzan's speed at the bottom of the swing, we can use the principle of conservation of mechanical energy. At the top of the swing, Tarzan has only potential energy, and at the bottom of the swing, he will have a combination of potential energy and kinetic energy.
(a) If Tarzan starts from rest, his initial speed is zero.
Using the conservation of mechanical energy equation:
Potential Energy at the top = Potential Energy at the bottom + Kinetic Energy at the bottom
mgh(top) = mgh(bottom) + (1/2)mv^2(bottom)
Where:
m = mass of Tarzan's body (assumed to be 1 kg in this example)
g = acceleration due to gravity (approximated as 9.8 m/s^2)
h(top) = height of the swing from the ground at the top (30.0 m)
h(bottom) = height of the swing from the ground at the bottom (0 m)
v(bottom) = velocity of Tarzan at the bottom (what we want to find)
Substituting the given values into the equation:
(1 kg)(9.8 m/s^2)(30.0 m) = (1 kg)(9.8 m/s^2)(0 m) + (1/2)(1 kg)(v^2(bottom))
294 J = 0 J + (1/2)v^2(bottom)
Rearranging the equation:
(1/2)v^2(bottom) = 294 J
v^2(bottom) = (294 J) * (2 / 1 kg)
v^2(bottom) = 588 m^2/s^2
Taking the square root of both sides of the equation:
v(bottom) = √(588 m^2/s^2)
v(bottom) ≈ 24.2 m/s
Therefore, if Tarzan starts from rest, his speed at the bottom of the swing is approximately 24.2 m/s.
(b) If Tarzan pushes off with a speed of 4.00 m/s, we can use the same conservation of mechanical energy equation:
Potential Energy at the top = Potential Energy at the bottom + Kinetic Energy at the bottom
mgh(top) + (1/2)mv^2(pushing off) = mgh(bottom) + (1/2)mv^2(bottom)
Substituting the given values into the equation:
(1 kg)(9.8 m/s^2)(30.0 m) + (1/2)(1 kg)(4.00 m/s)^2 = (1 kg)(9.8 m/s^2)(0 m) + (1/2)(1 kg)(v^2(bottom))
294 J + 8 J = 0 J + (1/2)v^2(bottom)
302 J = (1/2)v^2(bottom)
v^2(bottom) = (302 J) * (2 / 1 kg)
v^2(bottom) = 604 m^2/s^2
Taking the square root of both sides of the equation:
v(bottom) = √(604 m^2/s^2)
v(bottom) ≈ 24.6 m/s
Therefore, if Tarzan pushes off with a speed of 4.00 m/s, his speed at the bottom of the swing is approximately 24.6 m/s.
To calculate Tarzan's speed at the bottom of the swing, we can use the principle of conservation of mechanical energy.
(a) If Tarzan starts from rest, we can assume that his initial potential energy is converted to kinetic energy at the bottom of the swing. On his way down, Tarzan's potential energy decreases and his kinetic energy increases. At the bottom of the swing, when he is at the lowest point, all his potential energy is converted into kinetic energy.
The potential energy (PE) of an object at a certain height is given by the equation PE = mgh, where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
In this case, the initial height of Tarzan when he is at the highest point of the swing is 30.0 m. So, the initial potential energy is given by PE_i = mgh = m * 9.8 m/s^2 * 30.0 m.
At the bottom of the swing, the height is zero since Tarzan is at the lowest point. So, the final potential energy is zero (PE_f = 0).
According to the principle of conservation of mechanical energy, the initial potential energy is equal to the final kinetic energy at the bottom of the swing (KE_f). So, we can equate the two equations:
PE_i = KE_f
mgh = (1/2)mv^2
Since the mass (m) cancels out, we have:
gh = (1/2)v^2
Now we can substitute the given values into the equation. The gravitational acceleration (g) is approximately 9.8 m/s^2, and the height (h) is 30.0 m.
9.8 m/s^2 * 30.0 m = (1/2)v^2
294 m^2/s^2 = (1/2)v^2
Multiply by 2 to isolate v^2:
588 m^2/s^2 = v^2
Taking the square root of both sides, we find:
v = √588 m/s
So, if Tarzan starts from rest, his speed at the bottom of the swing is approximately 24.2 m/s.
(b) If Tarzan pushes off with an initial speed of 4.00 m/s, we need to consider both the initial potential energy and the initial kinetic energy.
The initial potential energy (PE_i) is the same as before, which is mgh.
The initial kinetic energy (KE_i) is given by the equation KE_i = (1/2)mv^2, where v is the initial velocity, which is 4.00 m/s.
Now we can equate the sum of initial potential energy and initial kinetic energy to the final kinetic energy at the bottom of the swing:
PE_i + KE_i = KE_f
mgh + (1/2)mv^2 = (1/2)mv_f^2
Again, the mass (m) cancels out:
gh + (1/2)v^2 = (1/2)v_f^2
Now we can substitute the given values. The gravitational acceleration (g) is approximately 9.8 m/s^2, the height (h) is 30.0 m, and the initial velocity (v) is 4.00 m/s.
9.8 m/s^2 * 30.0 m + (1/2)(4.00 m/s)^2 = (1/2)v_f^2
294 m^2/s^2 + 8.0 m^2/s^2 = (1/2)v_f^2
302 m^2/s^2 = (1/2)v_f^2
Multiply by 2 to isolate v_f^2:
604 m^2/s^2 = v_f^2
Taking the square root of both sides, we find:
v_f = √604 m/s
So, if Tarzan pushes off with a speed of 4.00 m/s, his speed at the bottom of the swing is approximately 24.6 m/s.