Lecture Notes: Kinetic Energy - University Physics Volume 1

Introduction to Kinetic Energy

• Kinetic Energy Definition:
  • Classical definition for non-relativistic speeds, defined as energy of motion.
  • Historically related to explaining collisions of elastic bodies (billiards example).
  • At relativistic speeds, requires different expression (covered in relativity).

Kinetic Energy Formula

• Basic Formula:

  • For a particle with mass ( m ) and speed ( v ): [ K = \frac{1}{2} m v^2 ]

• System of Particles:

  • Sum of kinetic energies of all particles in the system.

• Alternative Expression Using Momentum:

  • Momentum ( p = mv ): [ K = \frac{p^2}{2m} ]

• Units:

  • Kinetic energy: ( kg \cdot m^2/s^2 ) or Joules (J), consistent with work units.

Example Problems

Example 7.6: Kinetic Energy Calculations

(a) 80-kg athlete running at 10 m/s

• Calculation: ( K = \frac{1}{2} (80 \text{ kg})(10 \text{ m/s})^2 = 4.0 \text{ kJ} )

(b) Chicxulub crater impact asteroid

• Energy: ( 4.2 \times 10^{23} \text{ J} )
• Find mass: [ m = \frac{2K}{v^2} = \frac{2(4.2 \times 10^{23} \text{ J})}{(22 \text{ km/s})^2} = 1.7 \times 10^{15} \text{ kg} ]

(c) Thermal neutron

• Speed ( 2.2 \text{ km/s} )
• Neutron mass: ( 1.68 \times 10^{-27} \text{ kg} )
• Calculation: ( K = \frac{1}{2}(1.68 \times 10^{-27} \text{ kg})(2.2 \text{ km/s})^2 = 4.1 \times 10^{-21} \text{ J} )

Example 7.7: Kinetic Energy in Different Frames

1. Relative to Subway Car:

   • Person's speed: 1.50 m/s
   • ( K = \frac{1}{2}(75.0 \text{ kg})(1.50 \text{ m/s})^2 = 84.4 \text{ J} )

2. Relative to Tracks:

   • Train speed: 15.0 m/s
   • Person's speed relative to track: 13.5 m/s or 16.5 m/s
   • Kinetic energy: 6.83 kJ or 10.2 kJ

3. Relative to Frame Moving with Person:

   • ( v = 0 ), therefore ( K = 0 )

Frame of Reference

• Importance:

  • Kinetic energy is frame-dependent, varies with observer's frame.

• Practical Usage:

  • Choose frame to simplify analysis (e.g., external or internal frame).

Example 7.8: Special Names for Kinetic Energy

(a) Basketball horizontal kinetic energy:

• Speed: 7.5 m/s
  • Mass: 0.624 kg
  • ( K = \frac{1}{2}(0.624 \text{ kg})(7.5 \text{ m/s})^2 = 17.6 \text{ J} )

(b) Translational kinetic energy of molecules:

• Mass of molecule: 29 u
  • Speed: 500 m/s
  • Total kinetic energy: 1.80 kJ

(c) Comparative speed:

• Speed for equivalent energy: 76.0 m/s

Check Your Understanding

• Truck vs. Car:
  • Given same kinetic energy, truck (more mass) has less speed.
  • Given same speed, truck has greater kinetic energy.

Conclusion

• Types of Kinetic Energy:

  • Translational, rotational, thermal (internal), vibrational.
  • All represent energy due to motion.

• Practical Application:

  • Useful in physics problems involving motion and impact.