Collision Dynamics and Rocket Motion Notes

Overview of Collisions

• Types of Collisions:
  • Perfectly Inelastic: Objects stick together after the collision.
  • Partially Elastic: Objects do not stick but some kinetic energy is lost.
  • Perfectly Elastic: Total kinetic energy is conserved.

Key Equations

• Velocity Calculation (for inelastic collisions):

  • $v<em>1 = \frac{(m</em>1 - \epsilon m<em>2) u</em>1 + m<em>2 (1 + \epsilon) u</em>2}{m<em>1 + m</em>2}$
  • Where (\epsilon) is the coefficient of restitution, (u1) and (u2) are initial velocities, and (m1) and (m2) are masses of the colliding bodies.

• Kinetic Energy Equations:

  • Initial Kinetic Energy (KE): $KE = \frac{1}{2} m<em>1 v</em>1^2 + \frac{1}{2} m<em>2 v</em>2^2$
  • Final Kinetic Energy (KE'): $KE' = \frac{1}{2} m<em>1 (v</em>{1}')^2 + \frac{1}{2} m<em>2 (v</em>{2}')^2$
  • Change in Kinetic Energy: $\Delta KE = KE' - KE$

Examples of Calculations

• Example 1:

  • Given:
  • Masses: (m1 = 2.5 \text{ kg}, m2 = 1.5 \text{ kg}
  • Initial velocities: (u1 = 3 m/s, u2 = -0.5 m/s
  • Coefficient of restitution: (\epsilon = 0.8
  • Calculating new velocity:
  • Use the initial velocity equation to find (v1) and (v2):
  • The new velocity after the collision is calculated to be approx. (v_1' = 0.6375 m/s.

• Example 2:

  • Reversal in Perfectly Inelastic Collision:
  • Given mass: (m = 0.9 kg, object stopped after a collision; calculating velocity post-collision includes consideration of momentum conservation.
  • Initial state: Momentum before collision = Momentum after:
    • (0.9 v_g + 0.004 (400) = (0.9 + 0.004)(v') results in a final velocity of approximately (v' = -1.78 m/s.

Rocket Motion

• Rocket Equation (Tsiolkovsky's Equation):
  • (v{f} = v{e} ln \left(\frac{mi}{mf}\right)$$
  • Where (vf) is final velocity, (ve) is effective exhaust velocity, (mi) is initial mass, and (mf) is final mass after fuel has been expelled.