Section 3.8: Review Examples
This section provides review examples combining concepts of work, kinetic and potential energy, power, and efficiency. Each example is solved step by step to reinforce your problem-solving strategies.
Example 1: Kinetic Energy from Work
A 10 kg object is pushed with a force of 50 N along a horizontal frictionless surface for 5 m. Find its final speed if it starts from rest.
Work done: \( W = F \cdot d = 50 \times 5 = 250 \,\text{J} \).
This work becomes kinetic energy: \( KE = \tfrac{1}{2} m v^2 \).
\( 250 = 0.5 \times 10 \times v^2 \implies v^2 = 50 \implies v \approx 7.07 \,\text{m/s} \).
Example 2: Gravitational Potential Energy
A 2 kg mass is lifted vertically at constant speed through a height of 4 m. Calculate the work done and the increase in potential energy.
Work done against gravity: \( W = m g h = 2 \times 9.8 \times 4 = 78.4 \,\text{J} \).
Increase in potential energy: \( PE = 78.4 \,\text{J} \).
Since speed is constant, there is no change in kinetic energy.
Example 3: Power Output
A motor lifts a 150 kg crate vertically upward by 3 m in 6 seconds. Find the average power output.
Work done: \( W = m g h = 150 \times 9.8 \times 3 = 4410 \,\text{J} \).
Power: \( P = \tfrac{W}{t} = \tfrac{4410}{6} = 735 \,\text{W} \).
Example 4: Efficiency
A machine does 200 J of useful work but requires 250 J of input energy. Find its efficiency.
\( \eta = \tfrac{W_{out}}{W_{in}} \times 100\% = \tfrac{200}{250} \times 100\% = 80\% \).