7.2 Kinetic Energy

Kinetic energy is the energy of motion, calculated as half the product of mass and velocity squared. It's a key concept in understanding how objects move and interact, playing a crucial role in collisions, work, and power calculations.

Kinetic energy isn't fixed—it changes based on an object's reference frame and can convert to other energy forms. This ties into the broader principle of energy conservation, where total energy in a closed system remains constant, even as it shifts between different types.

Kinetic Energy

Kinetic energy calculation

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• Represents energy possessed by an object due to its motion
  • Calculated using formula $KE = \frac{1}{2}mv^2$, $m$ is mass and $v$ is velocity
• Depends on object's mass and velocity
  • Doubling mass doubles kinetic energy (2 kg object moving at 1 m/s has 2 J of KE)
  • Doubling velocity quadruples kinetic energy (1 kg object moving at 2 m/s has 4 J of KE)
• Also calculated using momentum ($p$)
  • Formula $KE = \frac{p^2}{2m}$, [p = mv](https://www.fiveableKeyTerm:p_=_mv)
  • 2 kg object with momentum of 4 kg⋅m/s has 4 J of KE
• Measured in joules (J) in SI units
  • 1 J = 1 kg⋅m²/s² (kinetic energy of 1 kg object moving at 1 m/s)
• Related to Newton's Second Law through force and acceleration
  • $F = ma$ can be used to determine change in kinetic energy over time

Kinetic energy in reference frames

• Relative quantity depends on frame of reference
  • Stationary ball on a moving train has KE relative to the ground
• Galilean relativity states physics laws are same in all inertial frames
  • Velocity is relative but velocity differences between frames are absolute
• Two frames A and B with relative velocity $[v_{AB}](https://www.fiveableKeyTerm:v_{AB})$
  • Object velocity $v_A$ in frame A is $v_B = v_A - v_{AB}$ in frame B
  • KE in frame B is $KE_B = \frac{1}{2}m(v_A - v_{AB})^2$
  • Ball thrown 10 m/s on train moving 20 m/s has KE of 45 J relative to ground
• Applies to objects in translational motion

Applications of kinetic energy

• Total energy conserved in closed system during collisions and interactions
  • KE can convert to potential, thermal, or other forms
• Elastic collisions conserve kinetic energy
  1. Ball bouncing on hard floor retains most of its KE after each bounce
  2. $KE_{initial} = KE_{final}$
• Inelastic collisions convert some KE to other forms
  1. Two clay balls stick together after colliding, losing KE to deformation
  2. $KE_{initial} > KE_{final}$
• Work-energy theorem relates work done to change in KE
  • $W = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$, $W$ is work, $v_i$ and $v_f$ are initial and final velocities
  • Pushing 1000 kg car from rest to 10 m/s requires 50,000 J of work
• Power is rate of doing work or transferring energy
  • $P = \frac{dW}{dt} = \frac{d}{dt}(\frac{1}{2}mv^2)$, $P$ is power, $t$ is time
  • 100 W motor can give 1 kg object KE of 50 J in 0.5 s

Energy Conservation and Mechanical Energy

• Kinetic energy is a component of mechanical energy
• In a closed system, total mechanical energy (kinetic + potential) remains constant
• Energy conservation principle applies to all forms of energy, including kinetic
• Work done on an object changes its mechanical energy