A 4.92E+2 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 +9.81E+4 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 7.90 m/s?
Do an energy balance.
Work done on balloon = (increases in KE) + (increase in gravitational potential energy)
Solve for the G.P.E change. You can use that number to compute the altitude.
It is OK to assume g is constant over the altitude range of a balloon.
To find the height above the surface of the earth at which the balloon has a speed of 7.90 m/s, we can use the principle of conservation of mechanical energy. The mechanical energy of the balloon is the sum of its kinetic energy and potential energy.
1. We can first calculate the initial potential energy of the balloon at the surface of the earth. The potential energy is given by the equation:
Potential Energy = Mass x Acceleration due to Gravity x Height
Since the balloon is at the surface of the earth and at rest, its initial velocity is zero. Therefore, the initial kinetic energy is zero.
The potential energy can be calculated as follows:
Initial Potential Energy = Mass x 9.81 m/s^2 x 0 (height at the surface of the earth)
2. Next, we can calculate the final kinetic energy of the balloon when it has a speed of 7.90 m/s. The kinetic energy is given by the equation:
Kinetic Energy = 0.5 x Mass x Velocity^2
Final Kinetic Energy = 0.5 x Mass x (7.90 m/s)^2
3. Since there are no other forms of energy involved in the problem, the work done on the balloon by the nonconservative wind and lift forces is equal to the change in mechanical energy. Therefore,
Work Done = Final Kinetic Energy - Initial Kinetic Energy
Work Done = 0.5 x Mass x (7.90 m/s)^2 - 0
4. Now, we can rearrange the equation to solve for the Mass of the balloon:
Mass = Work Done / (0.5 x (7.90 m/s)^2)
5. Once we have the Mass of the balloon, we can substitute it into the equation for potential energy to find the height:
Height = (Work Done) / (Mass x Acceleration due to Gravity)
Height = (9.81E+4 J) / (Mass x 9.81 m/s^2)
Note: In step 4, we can directly use the given mass of the balloon (4.92E+2 kg) if it is the same throughout the process. If there are changes in mass, we need to consider that.
By following these steps, we can calculate the height above the surface of the earth at which the balloon has a speed of 7.90 m/s.