Section 3.7: Energy Problem-Solving

In this section, we apply concepts of work, kinetic energy, potential energy, and efficiency to solve real-world physics problems. Understanding how to identify known and unknown quantities, select the appropriate energy equations, and solve systematically is essential.

Steps for Solving Energy Problems:
  1. Identify the system and type of energy involved (kinetic, potential, work done).
  2. Draw a diagram if necessary.
  3. List known quantities (mass, velocity, height, forces, etc.).
  4. Determine the unknown quantity to solve for.
  5. Apply the appropriate energy equations:
    • Kinetic Energy: \( KE = \frac{1}{2} m v^2 \)
    • Gravitational Potential Energy: \( PE = m g h \)
    • Work Done: \( W = F d \cos\theta \)
    • Conservation of Energy: \( KE_i + PE_i + W_{\text{non-conservative}} = KE_f + PE_f \)
  6. Solve algebraically and check units and reasonableness of answer.

Example 1

A 2 kg block slides down a frictionless incline of height 5 m. Find its speed at the bottom.

Use conservation of energy: \( PE_{\text{top}} = KE_{\text{bottom}} \)

\( m g h = \frac{1}{2} m v^2 \Rightarrow 2*9.8*5 = 0.5*2*v^2 \)

\( v^2 = 98 \Rightarrow v \approx 9.9 \, \text{m/s} \)

Example 2

A 5 kg box is pushed 4 m across a rough surface by a force of 20 N at an angle of 0°. If 30 J of energy is lost to friction, find the final kinetic energy of the box.

Work done by applied force: \( W = F d = 20*4 = 80 \, \text{J} \)

Energy lost to friction: 30 J

Final KE: \( KE_f = W - W_{\text{friction}} = 80 - 30 = 50 \, \text{J} \)

Practice Problems

  1. A 3 kg ball is dropped from 10 m. Find its speed just before hitting the ground.
  2. A 1000 kg car moves at 20 m/s. What is its kinetic energy?
  3. A 2 kg block is pushed up a frictionless incline of 3 m height. Find work done against gravity.
  4. An object slides down a frictionless slope, converting 200 J of potential energy into kinetic energy. Find final speed if mass is 4 kg.
  5. A motor does 500 J of work and 150 J is lost to friction. Find the increase in kinetic energy of the object.
  6. A 0.5 kg object falls from 8 m height onto a spring. Find velocity just before hitting spring. Ignore air resistance.
  7. A 10 kg box is pushed 5 m across a rough surface by 100 N force. Energy lost to friction is 40 J. Find final kinetic energy.
  8. A pendulum of mass 2 kg is raised 1.5 m. Find speed at lowest point.
  9. An object of mass 3 kg is thrown vertically upward at 10 m/s. Find maximum height reached.
  10. A sled moves down a 5 m slope with 80 J of initial potential energy. If friction dissipates 20 J, find speed at bottom.
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