Apply Momentum Conservation to Collisions - Physics

1. What is the impulse-momentum theorem for an object in 1D?

   $J = \Delta p = m(v-u)$. Impulse equals change in momentum, which is mass times velocity change.

2. What is the momentum conservation equation for a 1D collision in an isolated system?

   $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$. Total momentum before equals total momentum after in isolated systems.

3. Identify the sign convention rule for 1D momentum problems along a chosen axis.

   Choose + direction; velocities opposite it are negative. Consistent signs ensure momentum conservation equations work correctly.

4. What component equations express 2D momentum conservation for two objects?

   $p_{x,i}=p_{x,f}$ and $p_{y,i}=p_{y,f}$. Each component is independently conserved.

5. What is the 2D momentum conservation statement for an isolated collision?

   $\vec p_{\text{initial}}=\vec p_{\text{final}}$. Vector momentum is conserved in all directions.

6. What is the post-collision common velocity $v$ if two masses stick together in 1D?

   $v = \frac{m_1u_1 + m_2u_2}{m_1 + m_2}$. Total momentum divided by total mass gives common velocity.

7. Find $v_2$ if $m_1 = m_2$, $u_2 = 0$, and after collision $v_1 = 0$ (use momentum conservation).

   $v_2 = u_1$. Equal masses exchange velocities in this special case.

8. What value of $e$ corresponds to a perfectly inelastic collision (objects stick)?

   $e = 0$. Objects stick together with no relative motion after collision.

9. Find $v$ if $m_1=1\,\text{kg}$ at $u_1=8\,\text{m/s}$ sticks to $m_2=3\,\text{kg}$ at $u_2=0$.

   $v=2\,\text{m/s}$. $(1)(8)+(3)(0)=(1+3)v$ gives $v=2$ m/s.

10. Find $v$ if $m_1=3\,\text{kg}$ at $u_1=2\,\text{m/s}$ sticks to $m_2=3\,\text{kg}$ at $u_2=-2\,\text{m/s}$.

    $v=0\,\text{m/s}$. Equal and opposite momenta cancel when objects stick.

11. What value of $e$ corresponds to a perfectly inelastic collision (objects stick together)?

    $e=0$. Objects stick together with no relative motion after collision.

12. What condition must be true for total momentum to be conserved during a collision?

    $F_{\text{ext,net}} = 0$ (impulse from external forces is zero). No external forces means the system's total momentum stays constant.

13. What is the momentum-conservation equation for an isolated 1D collision of two objects?

    $m_1u_1+m_2u_2=m_1v_1+m_2v_2$. Total momentum before equals total momentum after for isolated systems.

14. Find the final velocity $v$ if $m_1 = 1\,\text{kg}$ at $u_1 = 4\,\text{m/s}$ hits $m_2 = 3\,\text{kg}$ at $u_2 = 0$ and they stick.

    $v = 1\,\frac{\text{m}}{\text{s}}$. $(1)(4) + (3)(0) = (1+3)v$ gives $v = 1$ m/s.

15. Identify the sign rule for 1D momentum problems when choosing a positive direction.

    Velocities opposite the positive direction are negative. Ensures correct vector addition in 1D problems.

16. What is the coefficient of restitution definition in 1D for two colliding objects?

    $e=\frac{v_2-v_1}{u_1-u_2}$. Measures relative separation speed over approach speed.

17. What is the momentum of a particle of mass $m$ moving at velocity $v$ in 1D?

    $p=mv$. Momentum equals mass times velocity in classical mechanics.

18. What value of $e$ corresponds to a perfectly elastic collision?

    $e=1$. Objects separate with same relative speed as approach.

19. Find the final velocity $v$ if $m_1 = 2\,\text{kg}$ at $u_1 = 3\,\text{m/s}$ hits $m_2 = 1\,\text{kg}$ at rest and they stick.

    $v = 2\,\text{m/s}$. $(2)(3) + (1)(0) = (2+1)v$ gives $v = 2$ m/s.

20. Find $v$ if $m_1=2\,\text{kg}$ at $u_1=3\,\text{m/s}$ sticks to $m_2=1\,\text{kg}$ at $u_2=0$.

    $v=2\,\text{m/s}$. $(2)(3)+(1)(0)=(2+1)v$ gives $v=2$ m/s.

21. Find the impulse on a $2\,\text{kg}$ cart that changes velocity from $-1\,\text{m/s}$ to $4\,\text{m/s}$.

    $J=10\,\text{N}\cdot\text{s}$. $J=m\Delta v=(2)(4-(-1))=10$ N·s.

22. Find the recoil speed $v_r$ of a $2\,\text{kg}$ gun if it fires a $0.010\,\text{kg}$ bullet at $400\,\text{m/s}$ from rest.

    $v_r = -2\,\text{m/s}$. $(2)(0) + (0.010)(0) = (2)v_r + (0.010)(400)$ gives $v_r = -2$ m/s.

23. Identify the correct 2D conservation equations if total initial momentum is only in $+x$ direction.

    $p_{x,i}=p_{x,f}$ and $0=p_{y,f}$. $p_{y,i}=0$ so $p_{y,f}=0$; x-momentum conserved separately.

24. What is the impulse-momentum theorem for one object in 1D?

    $J=\triangle p=m(v-u)$. Impulse equals change in momentum.

25. What additional conservation law applies to an elastic collision besides momentum?

    Kinetic energy is conserved: $K_i = K_f$. Elastic collisions conserve both momentum and kinetic energy.

26. Which quantity is always conserved in an isolated collision: momentum or kinetic energy?

    Momentum. KE only conserved in elastic collisions; momentum always conserved.

27. Find the system total momentum $p$ for $m_1 = 2\,\text{kg}$ at $+3\,\text{m/s}$ and $m_2 = 1\,\text{kg}$ at $-2\,\text{m/s}$.

    $p = 4\,\text{kg}\cdot\text{m/s}$. $(2)(3) + (1)(-2) = 6 - 2 = 4$ kg·m/s.

28. What is the kinetic energy of a mass $m$ moving at speed $v$?

    $K = \frac{1}{2}mv^2$. Energy of motion equals half mass times velocity squared.

29. State the formula for the common final velocity when two objects stick together in 1D.

    $v=\frac{m_1u_1+m_2u_2}{m_1+m_2}$. Total momentum divided by total mass gives common velocity.

30. What condition must be true for total momentum to be conserved during a collision?

    Net external impulse on the system is $0$. No external forces means momentum stays constant.

31. What is the momentum of a particle of mass $m$ moving at velocity $v$ in 1D?

    $p = mv$. Momentum equals mass times velocity in the chosen direction.

32. What value of $e$ corresponds to a perfectly elastic collision?

    $e = 1$. Objects separate at same relative speed they approached.

33. Find the impulse $J$ on a $0.50\,\text{kg}$ ball if it changes from $+4\,\text{m/s}$ to $-2\,\text{m/s}$.

    $J = -3\,\text{N}\cdot\text{s}$. $J = (0.50)[(-2) - (+4)] = -3$ N·s.

34. Find the final speed $v$ if a $0.20\,\text{kg}$ cart at $5\,\text{m/s}$ sticks to a $0.30\,\text{kg}$ cart at rest.

    $v = 2\,\text{m/s}$. $(0.20)(5) + (0.30)(0) = (0.50)v$ gives $v = 2$ m/s.

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