Question

elias serves a volleyball at a velocity of 16 m/s. the mass of the volleyball is 0.27 kg. what is the height of the volleyball above the gym floor if its total mechanical energy is 41.70 j? round to the nearest tenth. m

Explanation:

Step1: Recall the formula for total mechanical energy

Total mechanical energy \( E \) is the sum of kinetic energy \( KE \) and potential energy \( PE \). The formulas are \( KE = \frac{1}{2}mv^2 \) and \( PE = mgh \), so \( E = \frac{1}{2}mv^2 + mgh \).

Step2: Plug in the known values

We know \( E = 41.70 \, \text{J} \), \( m = 0.27 \, \text{kg} \), \( v = 16 \, \text{m/s} \), and \( g = 9.8 \, \text{m/s}^2 \). First, calculate the kinetic energy:
\( KE = \frac{1}{2} \times 0.27 \times (16)^2 \)
\( KE = 0.5 \times 0.27 \times 256 \)
\( KE = 0.135 \times 256 = 34.56 \, \text{J} \)

Step3: Solve for potential energy

Since \( E = KE + PE \), we can find \( PE = E - KE \).
\( PE = 41.70 - 34.56 = 7.14 \, \text{J} \)

Step4: Solve for height \( h \) from the potential energy formula

From \( PE = mgh \), we can rearrange to \( h = \frac{PE}{mg} \).
Plug in \( PE = 7.14 \, \text{J} \), \( m = 0.27 \, \text{kg} \), \( g = 9.8 \, \text{m/s}^2 \):
\( h = \frac{7.14}{0.27 \times 9.8} \)
First calculate the denominator: \( 0.27 \times 9.8 = 2.646 \)
Then \( h = \frac{7.14}{2.646} \approx 2.7 \, \text{m} \)

Answer:

\( 2.7 \)