A 5.00 x 102 kg hot air balloon takes off from rest at the surface of the earth. The buoyant force lifts the balloon straight up, performing 9.70 x 104 J of work on the balloon. At the instant the balloon's upward speed is 4.08 m/s, the balloon will be ____ m above the surface of the earth.
Excellent
Well, let me calculate that for you. But before I do, let's hope the balloon doesn't end up in the clouds, or it might get misty! Okay, here we go.
The work done on the balloon is equal to the change in its potential energy, given by the formula W = mgh, where W is the work done, m is the mass of the balloon, g is the acceleration due to gravity, and h is the height.
So, we have:
9.70 x 10^4 J = (5.00 x 10^2 kg)(9.8 m/s^2)(h)
Solving for h, we get:
h = 9.70 x 10^4 J / [(5.00 x 10^2 kg)(9.8 m/s^2)]
Calculating that out, we find that the balloon is approximately 1.98 meters above the surface of the earth.
Just high enough to avoid any low-flying birds and branches!
To find the height of the hot air balloon above the surface of the Earth, we can use the work-energy theorem.
The work-energy theorem states that the work done on an object is equal to its change in kinetic energy.
In this case, the work done by the buoyant force on the hot air balloon is equal to the change in its kinetic energy.
The work done on the balloon is given as 9.70 x 10^4 J.
The change in kinetic energy is equal to the final kinetic energy minus the initial kinetic energy.
We are given the initial speed of the balloon, which is 0 m/s, and the final speed, which is 4.08 m/s.
The final kinetic energy of the balloon is (1/2)mv^2, where m is the mass of the balloon and v is the final speed.
We are not given the mass of the balloon, but we are given the weight, which is equal to the mass multiplied by the acceleration due to gravity.
The weight of the balloon is given by the formula weight = mass x g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
In this case, the weight of the balloon is (5.00 x 10^2 kg) x (9.8 m/s^2) = 4.90 x 10^3 N.
The buoyant force is equal in magnitude to the weight of the balloon, so the work done by the buoyant force is also 4.90 x 10^3 J.
Therefore, the change in kinetic energy is 9.70 x 10^4 J - 4.90 x 10^3 J = 9.21 x 10^4 J.
Using the equation for kinetic energy, we can write:
(1/2)mv^2 = 9.21 x 10^4 J
We can rearrange this equation to solve for the mass of the balloon:
m = (2 x 9.21 x 10^4 J) / v^2
Substituting the given values for v, we get:
m = (2 x 9.21 x 10^4 J) / (4.08 m/s)^2
m = 4.51 x 10^3 kg
Now that we have the mass of the balloon, we can calculate the height above the surface of the Earth using the formula for potential energy:
potential energy = mass x gravity x height
The potential energy gained by the balloon is equal to the work done on the balloon by the buoyant force.
9.70 x 10^4 J = (4.51 x 10^3 kg) x (9.8 m/s^2) x height
height = (9.70 x 10^4 J) / ((4.51 x 10^3 kg) x (9.8 m/s^2))
height ≈ 2.15 m
Therefore, at the instant the balloon's upward speed is 4.08 m/s, the balloon will be approximately 2.15 m above the surface of the Earth.
To solve this problem, we can use the work-energy principle. The work done on the balloon by the buoyant force is equal to the change in its kinetic energy.
Given:
Mass of the balloon (m) = 5.00 x 10^2 kg
Work done on the balloon (W) = 9.70 x 10^4 J
Upward speed of the balloon (v) = 4.08 m/s
First, let's find the initial kinetic energy (K1) of the balloon when it is at rest:
K1 = 0.5 * m * v1^2
Since the balloon is at rest initially, v1 = 0.
Therefore, K1 = 0.5 * 5.00 x 10^2 kg * (0 m/s)^2 = 0 J.
Next, let's find the final kinetic energy (K2) of the balloon when its upward speed is 4.08 m/s:
K2 = 0.5 * m * v2^2
K2 = 0.5 * 5.00 x 10^2 kg * (4.08 m/s)^2 = 4.0872 x 10^3 J.
Now, we can use the work-energy principle to find the change in kinetic energy (∆K) of the balloon:
∆K = K2 - K1
∆K = 4.0872 x 10^3 J - 0 J = 4.0872 x 10^3 J.
Since work done (W) is equal to the change in kinetic energy (∆K), we have:
W = ∆K
9.70 x 10^4 J = 4.0872 x 10^3 J.
Finally, let's solve for the displacement (s) or the distance above the earth's surface:
s = W / (-mg)
where g is the acceleration due to gravity (9.8 m/s^2).
Plugging in the values, we can calculate s:
s = (9.70 x 10^4 J) / (-5.00 x 10^2 kg * 9.8 m/s^2) = -19.79591837 m.
Since displacement is a vector quantity, the negative sign indicates that the balloon is above the starting point (surface of the earth). Hence, the balloon will be approximately 19.8 m above the surface of the earth.
KE+PE=Total Work
((1/2)mv^2) + (mgh) = W
W = 9.7x10^4 J
v = 4.08 m/s
m = 500 kg
g = 9.8 m/s^2
Rearrage for h
h = (W - ((1/2)mv^2)) / mg
h = 18.9466m