Momentum & Collisions

Question 1 (10)

(a) From $\vec F = d\vec p/dt$, integrate over collision time: $\int_{t_1}^{t_2}\vec F\,dt = \int d\vec p = \vec p_2 - \vec p_1 = \Delta\vec p.$ Define impulse $\vec J \equiv \int_{t_1}^{t_2}\vec F\,dt$. Hence $\vec J = \Delta\vec p$. (1 integral setup, 1 evaluation, 1 definition)

(b) Particles 1, 2 exert forces on each other. By Newton's III: $\vec F_{12} = -\vec F_{21}$. For each: $\vec F_{21} = d\vec p_1/dt$, $\vec F_{12} = d\vec p_2/dt$ (only internal forces). Add: $\dfrac{d(\vec p_1+\vec p_2)}{dt} = \vec F_{21}+\vec F_{12} = 0.$ Therefore $\vec p_1 + \vec p_2 =$ constant. (1 III law, 1 each eqn, 1 conclusion)

(c) Momentum is conserved iff the net external force = 0 (vector sum). Internal forces occur in Newton-III pairs that cancel pairwise in the total sum $\sum \vec F_{ij}=0$, so they cannot alter total $\vec p$; only external forces can. (1 condition, 1 pairing, 1 why)

Question 2 (12)

(a) Momentum: $m_1u_1+m_2u_2 = m_1v_1+m_2v_2$. KE: $\tfrac12 m_1u_1^2+\tfrac12 m_2u_2^2 = \tfrac12 m_1v_1^2+\tfrac12 m_2v_2^2$. (2)

(b) Rearrange: $m_1(u_1-v_1)=m_2(v_2-u_2)$ …(i) $m_1(u_1^2-v_1^2)=m_2(v_2^2-u_2^2)$ …(ii) Divide (ii) by (i): $u_1+v_1 = v_2+u_2 \Rightarrow u_1-u_2 = v_2-v_1$ (relative velocity reverses). (3) Solve with (i): $v_1=\frac{(m_1-m_2)u_1+2m_2u_2}{m_1+m_2},\quad v_2=\frac{(m_2-m_1)u_2+2m_1u_1}{m_1+m_2}.$ (3)

(c) $m_1=2,m_2=4,u_1=6,u_2=0$: $v_1=\dfrac{(2-4)6}{6}=\dfrac{-12}{6}=-2.0\text{ m/s}$. $v_2=\dfrac{2\cdot2\cdot6}{6}=\dfrac{24}{6}=4.0\text{ m/s}$. (2+2)

Question 3 (10)

(a) $e = \dfrac{v_2-v_1}{u_1-u_2}$ = (relative speed of separation)/(relative speed of approach). Elastic $e=1$; perfectly inelastic $e=0$. (2)

(b) Common velocity $v=\dfrac{m_1u_1}{m_1+m_2}$. Initial KE $=\tfrac12 m_1u_1^2$. Final $=\tfrac12(m_1+m_2)v^2 = \tfrac12\dfrac{m_1^2u_1^2}{m_1+m_2}$. $\frac{\Delta KE}{KE_i}=1-\frac{m_1}{m_1+m_2}=\frac{m_2}{m_1+m_2}.$ (2 velocity, 2 energies, 1 fraction)

(c) $v=\dfrac{3\cdot8}{4}=6.0\text{ m/s}$. Fractional loss $=\dfrac{m_2}{m_1+m_2}=\dfrac{1}{4}=0.25$ (25%). (2+1)

Question 4 (10)

(a) $\vec r_{\text{CM}} = \dfrac{\sum m_i \vec r_i}{\sum m_i}$ (or $\dfrac1M\int \vec r\,dm$). (2)

(b) Semicircular arc, radius $R$, linear density $\lambda = M/(\pi R)$. Symmetry ⇒ CM on the axis (call $y$). Element at angle $\theta$: $dm=\lambda R\,d\theta$, $y=R\sin\theta$, $\theta\in[0,\pi]$. $y_{\text{CM}}=\frac{1}{M}\int_0^\pi R\sin\theta\,\lambda R\,d\theta=\frac{\lambda R^2}{M}\int_0^\pi\sin\theta\,d\theta=\frac{\lambda R^2}{M}\,[{-\cos\theta}]_0^\pi=\frac{\lambda R^2}{M}(2).$ With $\lambda R = M/\pi$: $y_{\text{CM}} = \dfrac{2R}{\pi}$. (2 setup, 2 integral, 2 result)

(c) Semicircular lamina: $y_{\text{CM}} = \dfrac{4R}{3\pi}$. (2)

Question 5 (10)

(a) $M\vec r_{\text{CM}}=\sum m_i\vec r_i$. Differentiate twice: $M\vec a_{\text{CM}}=\sum m_i\vec a_i=\sum \vec F_i$. Internal forces cancel (III law) ⇒ $\sum\vec F_i = \vec F_{\text{ext}}$. Hence $\vec F_{\text{ext}}=M\vec a_{\text{CM}}$. (1+1+1+1)

(b) Equal masses $m$. Momentum: $\vec u = \vec v_1+\vec v_2$. KE (÷$\tfrac12 m$): $u^2 = v_1^2+v_2^2$. Square the vector eqn: $u^2=\vec u\cdot\vec u=(\vec v_1+\vec v_2)^2=v_1^2+v_2^2+2\vec v_1\cdot\vec v_2$. Comparing: $2\vec v_1\cdot\vec v_2 = 0 \Rightarrow \vec v_1\perp\vec v_2$ (provided both nonzero). (2 momentum, 2 energy, 2 dot-product conclusion)

Question 6 (8)

(a) In dt, mass $-dm$ (i.e. $dm<0$) ejected at exhaust speed $v_{\text{ex}}$ relative to rocket. Momentum conservation (no external force): $m\,dv = -v_{\text{ex}}\,dm.$ $\int_0^{\Delta v}dv = -v_{\text{ex}}\int_{m_0}^{m_f}\frac{dm}{m}\Rightarrow \Delta v = v_{\text{ex}}\ln\frac{m_0}{m_f}.$ (2 momentum eqn, 2 integrate, 1 result)

(b)

import math
def delta_v(m0, mf, vex):
    return vex * math.log(m0/mf)

$\Delta v = 2500\ln(1000/400)=2500\ln(2.5)=2500(0.9163)=2290.7\text{ m/s}$. (2 code, 1 value)

[
{"claim":"Elastic collision v1 = -2 m/s for m1=2,m2=4,u1=6,u2=0","code":"m1,m2,u1,u2=2,4,6,0\nv1=((m1-m2)*u1+2*m2*u2)/(m1+m2)\nresult = v1==-2"},
{"claim":"Elastic collision v2 = 4 m/s","code":"m1,m2,u1,u2=2,4,6,0\nv2=((m2-m1)*u2+2*m1*u1)/(m1+m2)\nresult = v2==4"},
{"claim":"Inelastic common velocity 6 m/s and fractional KE loss 0.25","code":"m1,m2,u1=3,1,8\nv=m1*u1/(m1+m2)\nfrac=m2/(m1+m2)\nresult = (v==6) and (frac==Rational(1,4))"},
{"claim":"Semicircular arc CM = 2R/pi","code":"R,th=symbols('R theta',positive=True)\ny=integrate(R*sin(th)*R,(th,0,pi))/(pi*R)\nresult = simplify(y-2*R/pi)==0"},
{"claim":"Rocket delta_v ~2290.7 m/s","code":"dv=2500*log(Rational(1000,400))\nresult = abs(float(dv)-2290.73)<0.5"}
]