Jane, looking for Tarzan, is running at top speed (6.2 m/s) and grabs a vine hanging vertically from a tall tree in the jungle. How high can she swing upward?
You just need to calculate Jane's kinectic energy before she grabs the vine (.5mv^2). This will be equal to her gravitational potential energy (mg*height) at the top of her swing.
.5mv^2=mgh
-cancel the mass on both sides
.5v^2=gh
19.22=9.8h
h=1.96
@Peter What about the non-conservative work lost with the tension force?
To determine how high Jane can swing upward, we can use the principles of conservation of mechanical energy. The initial mechanical energy of Jane when she grabs the vine will be equal to the potential energy she gains during the swing.
The formula for potential energy is:
Potential Energy (PE) = mass (m) × gravitational acceleration (g) × height (h)
Jane's mass does not affect her upward swing, so we can focus on the height she can reach.
We can rearrange the formula to solve for height (h):
h = PE / (m × g)
Given:
Speed of Jane (v) = 6.2 m/s
Gravitational acceleration (g) = 9.8 m/s^2 (approximately)
First, we need to calculate Jane's initial potential energy (PE).
The formula for potential energy is:
PE = mass (m) × gravitational acceleration (g) × height (h)
Since we don't have the mass of Jane, we can assume a value of 1 kg for simplicity.
Let's calculate Jane's initial potential energy (PE):
PE = 1 kg × 9.8 m/s^2 × h
Jane's initial kinetic energy (KE) can be calculated using the formula:
KE = (1/2) × mass (m) × velocity^2 (v^2)
Substituting the given values, we get:
KE = (1/2) × 1 kg × (6.2 m/s)^2
Now, we can use conservation of mechanical energy to find the height Jane can swing upward. The initial mechanical energy (E_i) is equal to the initial kinetic energy (KE) Jane grabs the vine:
E_i = KE + PE
E_i = (1/2) × 1 kg × (6.2 m/s)^2 + 1 kg × 9.8 m/s^2 × h
Jane's final mechanical energy (E_f) at the highest point of her swing is equal to her potential energy (PE) at that point (as her kinetic energy will be zero):
E_f = PE = 1 kg × 9.8 m/s^2 × h
As energy is conserved, we can equate E_i and E_f:
(1/2) × 1 kg × (6.2 m/s)^2 + 1 kg × 9.8 m/s^2 × h = 1 kg × 9.8 m/s^2 × h
Now, we can solve for h:
(1/2) × 6.2^2 + 9.8 × h = 9.8 × h
(1/2) × 6.2^2 = 9.8 × h - 9.8 × h
(1/2) × 6.2^2 = 0
Therefore, the height Jane can swing upward is 0 meters.
To determine how high Jane can swing upward, we need to use the conservation of mechanical energy. The mechanical energy at the bottom (when she grabs the vine) will be equal to the mechanical energy at the highest point of her swing. This can be expressed as:
mgh = 1/2 mv^2
where m is Jane's mass, g is the acceleration due to gravity, h is the height she swings upward, and v is Jane's velocity.
Since Jane's mass is not provided, we can assume it cancels out when comparing the mechanical energies. We can also neglect air resistance in this calculation.
First, let's determine the value of g. The acceleration due to gravity on Earth is approximately 9.8 m/s^2.
Now, we can plug in the values and solve for h:
mgh = 1/2 mv^2
gh = 1/2 v^2
h = (1/2 v^2) / g
h = (1/2 * (6.2 m/s)^2) / 9.8 m/s^2
h ≈ 1.6071 meters
Therefore, Jane can swing upward to a height of approximately 1.6071 meters.