PHSX 220 Physics I (w/ Calculus) · Unit 3: Momentum, rotation, gravitation · on Exam 3
Ch 9.4–9.6 Collisions and center of mass
Study notes for Collisions and center of mass in PHSX 220 Physics I (w/ Calculus) at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- All collisions in an isolated system conserve momentum. Elastic collisions also conserve kinetic energy; inelastic ones lose some KE (to heat, sound, deformation); perfectly inelastic ones stick together and lose the most KE possible.
- Perfectly inelastic: $v_f = \dfrac{m_1v_1 + m_2v_2}{m_1+m_2}$. Elastic with target at rest: $v_{1f} = \dfrac{m_1 - m_2}{m_1+m_2}v_{1i}$, $v_{2f} = \dfrac{2m_1}{m_1+m_2}v_{1i}$. Equal masses swap velocities.
- 2D collisions: conserve $p_x$ and $p_y$ separately. Draw before and after pictures with angles.
- Center of mass $x_{cm} = \dfrac{\sum m_ix_i}{\sum m_i}$ moves as if all mass were there and all external forces acted on it. Internal forces (explosions, collisions) never change the center-of-mass motion.
Key formulas
Perfectly inelastic
$$v_f = \frac{m_1v_{1i} + m_2v_{2i}}{m_1+m_2}$$
Elastic, target at rest
$$v_{1f} = \frac{m_1-m_2}{m_1+m_2}v_{1i},\qquad v_{2f} = \frac{2m_1}{m_1+m_2}v_{1i}$$
Center of mass
$$\vec r_{cm} = \frac{\sum m_i\vec r_i}{\sum m_i},\qquad \vec v_{cm} = \frac{\vec p_{\text{total}}}{M}$$
Worked example
A 2.0 kg cart at 3.0 m/s hits a 1.0 kg cart at rest and they stick. Find the final speed and the KE lost.
Show the solution
$v_f = 2(3)/3 = 2.0$ m/s. $K_i = \tfrac12(2)(9) = 9$ J; $K_f = \tfrac12(3)(4) = 6$ J; lost $3$ J (33%).
Common mistakes
- Assuming kinetic energy is conserved in every collision.
- Forgetting momentum is a vector in 2D problems.
- Using speeds without signs for head-on collisions.
Exam tipBallistic pendulum problems chain two tools: momentum conservation for the (inelastic) impact, then energy conservation for the swing. Never energy across the impact.
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