The change in mechanical energy of the ball-earth system due to air drag is a loss of 4.384 J. This was calculated using the initial and final kinetic and potential energies, accounting for the change in both height and speed.
To determine the change in mechanical energy of the ball-earth system due to air drag, we'll calculate the mechanical energy at two points during its motion and take the difference. Mechanical energy is the sum of kinetic energy (KE) and gravitational potential energy (PE).
Initially, the ball has kinetic energy (KE_initial) given by \(\frac{1}{2}mv²\) and gravitational potential energy (PE_initial) given by mgh, where m is the mass of the ball, v is the velocity, g is the acceleration due to gravity (9.8 m/s2), and h is the height above the ground.
KE_initial = \(\{1}{2} · 0.068 kg · (15.9 m/s)²\) = 8.61 J
PE_initial = 0.068 kg · 9.8 m/s2 · 1.03 m = 0.684 J
After the ball has risen to 1.53 m, its kinetic energy (KE_final) and potential energy (PE_final) are given by:
KE_final = \(\{1}{2} · 0.068 kg · (10.7 m/s)²\) = 3.89 J
PE_final = 0.068 kg · 9.8 m/s2 · 1.53 m = 1.02 J
The total mechanical energy initially (ME_initial) and finally (ME_final) are:
ME_initial = KE_initial + PE_initial = 8.61 J + 0.684 J = 9.294 J
ME_final = KE_final + PE_final = 3.89 J + 1.02 J = 4.91 J
The change in mechanical energy due to air drag \(\Δ ME\) is:
\(\ΔME\) = ME_final - ME_initial = 4.91 J - 9.294 J = -4.384 J
The negative sign indicates a loss of mechanical energy, which is the work done by air drag.