To find the velocity of the hot-air balloon, we use the formula for velocity derived from kinetic energy:
[tex]\[ V = \sqrt{\frac{2KE}{m}} \][/tex]
Where:
- [tex]\( V \)[/tex] is the velocity of the balloon.
- [tex]\( KE \)[/tex] is the kinetic energy of the balloon.
- [tex]\( m \)[/tex] is the mass of the balloon.
Given data:
- The kinetic energy ([tex]\( KE \)[/tex]) is 3,200 joules.
- The mass ([tex]\( m \)[/tex]) is 400 kilograms.
Step-by-step, the calculation proceeds as follows:
1. Substitute the given values into the formula:
[tex]\[ V = \sqrt{\frac{2 \times 3200 \, \text{J}}{400 \, \text{kg}}} \][/tex]
2. Calculate the numerator of the fraction:
[tex]\[ 2 \times 3200 = 6400 \, \text{J} \][/tex]
3. Substitute this back into the formula:
[tex]\[ V = \sqrt{\frac{6400 \, \text{J}}{400 \, \text{kg}}} \][/tex]
4. Divide the numerator by the denominator:
[tex]\[ \frac{6400}{400} = 16 \, \text{m}^2/\text{s}^2 \][/tex]
5. Take the square root of the result:
[tex]\[ \sqrt{16 \, \text{m}^2/\text{s}^2} = 4.0 \, \text{m/s} \][/tex]
Therefore, the velocity of the balloon is [tex]\( 4.0 \)[/tex] meters per second.
