Final answer:
- (a) Tarzan's speed at the bottom of the swing if he starts from rest is approximately 23.9 m/s
- (b) His speed at the bottom of the swing if he pushes off with a speed of 6.00 m/s is approximately 23.2 m/s.
Step-by-step explanation:
a) 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, all of Tarzan's initial potential energy will be converted into kinetic energy at the bottom of the swing, assuming no energy losses due to friction or air resistance.
First, let's find Tarzan's initial potential energy. The potential energy (PE) is given by the formula:
PE = m * g * h
where m is Tarzan's mass, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the vertical height.
Since Tarzan starts from rest, his initial speed is zero, and therefore the initial height is the same as the length of the vine, which is 29.2 m.
PE = m * g * h
PE = m * 9.8 * 29.2
Next, let's find Tarzan's final kinetic energy. The kinetic energy (KE) is given by the formula:
KE = (1/2) * m * v²
where v is the final speed. At the bottom of the swing, all of the initial potential energy is converted into kinetic energy:
PE = KE
m * 9.8 * 29.2 = (1/2) * m * v²
We can cancel out the mass, m, from both sides of the equation:
9.8 * 29.2 = (1/2) * v²
Now, let's solve for v:
- v² = (2 * 9.8 * 29.2) / 1
- v² = 2 * 9.8 * 29.2
- v² = 571.04
Taking the square root of both sides, we find:
v = √571.04
v ≈ 23.9 m/s
Therefore, Tarzan's speed at the bottom of the swing, starting from rest, is approximately 23.9 m/s.
b) To find Tarzan's speed at the bottom of the swing when he pushes off with a speed of 6.00 m/s, we need to consider the conservation of mechanical energy and the effects of both the initial speed and the initial potential energy.
First, let's find Tarzan's initial potential energy. The potential energy (PE) is given by the formula:
PE = m * g * h
where m is Tarzan's mass, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the vertical height.
Since Tarzan starts from rest, his initial speed is zero, and therefore the initial height is the same as the length of the vine, which is 29.2 m.
PE = m * g * h
PE = m * 9.8 * 29.2
Next, let's consider Tarzan's initial kinetic energy. The kinetic energy (KE) is given by the formula:
KE = (1/2) * m * v²
where v is Tarzan's initial speed, which is given as 6.00 m/s.
KE = (1/2) * m * v²
KE = (1/2) * m * (6.00)²
Now, let's find Tarzan's final kinetic energy at the bottom of the swing. At the bottom, Tarzan's initial potential energy is completely converted into kinetic energy.
PE = KE
m * 9.8 * 29.2 = (1/2) * m * (6.00)² + KE
We can rearrange the equation to solve for KE:
KE = m * 9.8 * 29.2 - (1/2) * m * (6.00)²
Now, let's solve for KE:
KE = m * 9.8 * 29.2 - (1/2) * m * (6.00)²
Since Tarzan's speed at the bottom of the swing is the same as the final speed, we can use the kinetic energy formula to find the speed:
KE = (1/2) * m * v²
Let's solve for v:
- v² = (2 * KE) / m
- v² = (2 * (m * 9.8 * 29.2 - (1/2) * m * (6.00)²)) / m
- v² = (2 * (9.8 * 29.2 - (1/2) * (6.00)²))
Now, let's solve for v:
v = √((2 * (9.8 * 29.2 - (1/2) * (6.00)²)))
v ≈ 23.2 m/s
Therefore, Tarzan's speed at the bottom of the swing, when he pushes off with a speed of 6.00 m/s, is approximately 23.2 m/s.
Your question is incomplete, but most probably the full question was:
Tarzan swings on a 29.2-m-long vine initially inclined at an angle of 38.0° with 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 pushes off with a speed of 6.00 m/s?