## Given Data - Mass of basketball, \( m = .5 \) kg - Height at peak, \( h = 4 \) m - Vertical velocity at peak, \( v = \) m/s - Acceleration due to gravity, \( g = 9.8 \) m/s² --- ### Step 1: Kinetic Energy at the Peak **Formula:** \[ KE = \frac{1}{2}mv^2 \] **Substitute values:** \[ KE = \frac{1}{2} \times .5 \times ()^2 = \text{ J} \] **Explanation:** At the peak, the vertical velocity is zero, so the kinetic energy is zero. --- ### Step 2: Gravitational Potential Energy at the Peak **Formula:** \[ PE = mgh \] **Substitute values:** \[ PE = .5 \times 9.8 \times 4 = 19.6 \text{ J} \] **Explanation:** At the peak, all the mechanical energy is stored as gravitational potential energy. --- ### Step 3: Total Mechanical Energy at the Peak **Formula:** \[ \text{Total Mechanical Energy} = KE + PE \] \[ = + 19.6 = 19.6 \text{ J} \] **Explanation:** The total mechanical energy is the sum of kinetic and potential energy at the peak. --- ## **Final Answers** a) **Kinetic energy at the peak**: ** J** b) **Gravitational potential energy at the peak**: **19.6 J** c) **Total mechanical energy at the peak**: **19.6 J**