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=(m</em>1−ϵm<em>2)u</em>1+m<em>2(1+ϵ)u</em>2m<em>1+m</em>2v<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=12m<em>1v</em>12+12m<em>2v</em>22KE = \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′=12m<em>1(v</em>1′)2+12m<em>2(v</em>2′)2KE' = \frac{1}{2} m<em>1 (v</em>{1}')^2 + \frac{1}{2} m<em>2 (v</em>{2}')^2
    • Change in Kinetic Energy: ΔKE=KE′−KE\Delta KE = KE' - KE

Examples of Calculations

  • Example 1:

    • Given:
    • Masses: (m1 = 2.5 \text{ kg}, m2 = 1.5 \text{ kg}</li><li>Initialvelocities:(u<em>1=3m/s,u</em>2=−0.5m/s</li> <li>Initial velocities: (u<em>1 = 3 m/s, u</em>2 = -0.5 m/s
    • Coefficient of restitution: (\epsilon = 0.8</li><li>Calculatingnewvelocity:</li><li>Usetheinitialvelocityequationtofind(v<em>1)and(v</em>2):</li><li>Thenewvelocityafterthecollisioniscalculatedtobeapprox.(v1′=0.6375m/s</li> <li>Calculating new velocity:</li> <li>Use the initial velocity equation to find (v<em>1) and (v</em>2): </li> <li>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,objectstoppedafteracollision;calculatingvelocitypost−collisionincludesconsiderationofmomentumconservation.</li><li>Initialstate:Momentumbeforecollision=Momentumafter:<ul><li>(0.9vg+0.004(400)=(0.9+0.004)(v′)resultsinafinalvelocityofapproximately(v′=−1.78m/s, object stopped after a collision; calculating velocity post-collision includes consideration of momentum conservation.</li> <li>Initial state: Momentum before collision = Momentum after: <ul> <li>(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.