A 490 kg hot-air balloon takes off from rest at the surface of the earth. The nonconservative wind and lift forces take the balloon up, doing 97000 J of work on the balloon in the process. At what height above the surface of the earth does the balloon have a speed of 7.40 m/s?
m g = weight = 490*9.81
so potential energy gained at height h = 4806 h Joules
work in =9700 = 4806 h + (1/2)(490) (7.4)^2
typo
work in =97,000 = 4806 h + (1/2)(490) (7.4)^2
To solve this problem, we will use the principle of conservation of energy. The initial potential energy of the balloon equals the work done on it plus its initial kinetic energy.
Let's break down the steps to finding the height above the surface of the earth:
Step 1: Find the initial potential energy of the balloon.
The initial potential energy (U_i) is given by the equation:
U_i = m * g * h
where m is the mass of the balloon (490 kg), g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the initial height above the surface of the earth (we will assume it to be zero).
U_i = 490 kg * 9.8 m/s^2 * 0 = 0 J
Since the balloon starts at rest at the surface of the earth, its initial kinetic energy (K_i) is zero.
K_i = 0.5 * m * v_i^2
where v_i is the initial velocity of the balloon (which equals zero in this case).
K_i = 0.5 * 490 kg * (0 m/s)^2 = 0 J
Step 2: Find the final potential energy of the balloon.
The final potential energy (U_f) can be found using the same equation as before, but with the final height (h_f). We will assume that the height above the surface of the earth when the balloon has a speed of 7.40 m/s is h_f.
U_f = m * g * h_f
Step 3: Find the final kinetic energy of the balloon.
The final kinetic energy (K_f) can be found using the equation:
K_f = 0.5 * m * v_f^2
where v_f is the final velocity of the balloon (7.40 m/s).
Step 4: Apply the principle of conservation of energy.
According to the conservation of energy principle, the initial potential energy plus the initial kinetic energy should equal the final potential energy plus the final kinetic energy.
U_i + K_i = U_f + K_f
0 J + 0 J = m * g * h_f + 0.5 * m * v_f^2
Simplifying the equation, we get:
0 = m * g * h_f + 0.5 * m * v_f^2
Substituting the known values:
0 = 490 kg * 9.8 m/s^2 * h_f + 0.5 * 490 kg * (7.40 m/s)^2
Solving for h_f, we have:
0 = 490 kg * 9.8 m/s^2 * h_f + 0.5 * 490 kg * (7.40 m/s)^2
Rearranging the equation, we get:
h_f = (0.5 * 490 kg * (7.40 m/s)^2) / (490 kg * 9.8 m/s^2)
h_f = (0.5 * 7.40 m/s)^2 / (9.8 m/s^2)
h_f = (0.5 * 7.40^2) / 9.8
h_f = 2.70 m
Therefore, the balloon is approximately 2.70 meters above the surface of the earth when its speed is 7.40 m/s.