Problem Statement
Solve the momentum/collision problem: Estimate the average number of collisions per second for a nitrogen molecule at 300 K and 1 atm. ($d = 3.7 \times 10^{-10}$ m, $M = 28 \times 10^{-3}$ kg/mol) $d = 3.7 \times 10^{-10}$ m $T = 300$ K, $P = 1.013 \times 10^5$ Pa $M = 28 \times 10^{-3}$ kg/mol Collision frequency $z$ (collisions per se
Given Information
- Masses $m_1$, $m_2$ and initial velocities $u_1$, $u_2$ as given
- Type of collision: elastic (KE conserved), perfectly inelastic (objects stick), or partially inelastic
Physical Concepts & Formulas
Linear momentum $\vec{p} = m\vec{v}$ is conserved whenever the net external force on the system is zero. In collisions, the collision forces are internal and huge but brief — the impulse-momentum theorem shows that external forces (gravity, friction) contribute negligible impulse during the short collision time. For elastic collisions, kinetic energy is also conserved, giving two equations for two unknowns. For perfectly inelastic collisions, objects merge and momentum alone governs the outcome.
- $m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$ — momentum conservation
- Elastic: $\frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2$
- Elastic result: $v_1 = \dfrac{(m_1-m_2)u_1+2m_2 u_2}{m_1+m_2}$
- Perfectly inelastic: $(m_1+m_2)v_f = m_1 u_1 + m_2 u_2$
Step-by-Step Solution
Step 1 — Identify type: Elastic, inelastic, or perfectly inelastic.
Step 2 — Write conservation equations:
$$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$$
Step 3 — For elastic collisions, add energy equation or use relative velocity relation: $(u_1 – u_2) = -(v_1-v_2)$.
Step 4 — Solve simultaneously for $v_1$ and $v_2$.
Worked Calculation
Substituting all values with units:
$$v_f = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}\quad\text{(perfectly inelastic)}$$
Answer
$$\boxed{v_f = \dfrac{m_1 u_1 + m_2 u_2}{m_1+m_2}}$$
Physical Interpretation
In a perfectly inelastic collision the kinetic energy lost $\Delta KE = \frac{1}{2}\frac{m_1 m_2}{m_1+m_2}(u_1-u_2)^2$ is converted to heat, sound, and deformation. When equal masses collide elastically, they exchange velocities — as seen in Newton’s cradle. When a very light ball hits a very heavy stationary wall, it bounces back at nearly the same speed — confirming the elastic formula.
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