To find the total mechanical energy of the ball at the moment it is tossed upward, we need to calculate both its kinetic energy (KE) and its gravitational potential energy (PE).
- Calculate the Kinetic Energy (KE): The formula for kinetic energy is:
\[ KE = \frac{1}{2}mv^2 \]
where:
- \( m \) is the mass of the ball (0.135 kg),
- \( v \) is the initial speed of the ball (15.0 m/s).
Substituting the values:
\[ KE = \frac{1}{2} \times 0.135 , \text{kg} \times (15.0 , \text{m/s})^2 \] \[ KE = \frac{1}{2} \times 0.135 \times 225 \] \[ KE = \frac{1}{2} \times 30.375 \] \[ KE = 15.1875 , \text{J} \]
- Calculate the Gravitational Potential Energy (PE): The formula for gravitational potential energy is:
\[ PE = mgh \]
where:
- \( h \) is the height above the ground (1.20 m),
- \( g \) is the acceleration due to gravity (approximately \( 9.81 , \text{m/s}^2 \)).
Substituting the values:
\[ PE = 0.135 , \text{kg} \times 9.81 , \text{m/s}^2 \times 1.20 , \text{m} \] \[ PE = 0.135 \times 11.772 \] \[ PE \approx 1.588 , \text{J} \]
- Total Mechanical Energy (E): The total mechanical energy is the sum of kinetic and potential energy:
\[ E = KE + PE \]
Substituting the values we calculated:
\[ E = 15.1875 , \text{J} + 1.588 , \text{J} \] \[ E \approx 16.7755 , \text{J} \]
Therefore, the ball’s total mechanical energy at the moment it is tossed upward is approximately 16.78 J.
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