Respuesta :
Answer:
(a) [tex]v_1 = m_2\sqrt{\frac{2G}{d(m_1+m_2)} }[/tex]
[tex]v_2=m_1\sqrt{\frac{2G}{d(m_1+m_2)} }[/tex]
(b) Kinetic Energy of planet with mass m₁, is KE₁ = 1.068×10³² J
Kinetic Energy of planet with mass m₂, KE₂ = 2.6696×10³¹ J
Explanation:
Here we have when their distance is d apart
[tex]F_{1} = F_{2} =G\frac{m_{1}m_{2}}{d^{2}}[/tex]
Energy is given by
[tex]Energy \,of \,attraction = -G\frac{m_{1}m_{2}}{d}}+\frac{1}{2} m_{1} v^2_1+ \frac{1}{2} m_{2} v^2_2[/tex]
Conservation of linear momentum gives
m₁·v₁ = m₂·v₂
From which
v₂ = m₁·v₁/m₂
At equilibrium, we have;
[tex]G\frac{m_{1}m_{2}}{d}} = \frac{1}{2} m_{1} v^2_1+ \frac{1}{2} m_{2} v^2_2[/tex] which gives
[tex]2G{m_{1}m_{2}}= d m_{1} v^2_1+ dm_{2} (\frac{m_1}{m_2}v_1)^2= dv^2_1(m_1+(\frac{m_1}{m_2} )^2)[/tex]
multiplying both sides by m₂/m₁, we have
[tex]2Gm^2_{2}}= dv^2_1 m_2+dm_1v^2_1 =dv^2_1( m_2+m_1)[/tex]
Such that v₁ = [tex]\sqrt{\frac{2Gm^2_2}{d(m_1+m_2)} }[/tex]
[tex]v_1 = m_2\sqrt{\frac{2G}{d(m_1+m_2)} }[/tex]
Similarly, with v₁ = m₂·v₂/m₁, we have
[tex]G\frac{m_{1}m_{2}}{d}} = \frac{1}{2} m_{1} v^2_1+ \frac{1}{2} m_{2} v^2_2\Rightarrow 2G{m_{1}m_{2}}= dm_{1} (\frac{m_2}{m_1}v_1)^2 +d m_{2} v^2_2= dv^2_2(m_2+(\frac{m_2}{m_1} )^2)[/tex]
From which we have;
[tex]2G{m^2_{1}}= dm_{2} v_2^2 +d m_{1} v^2_2[/tex] and
[tex]v_2=m_1\sqrt{\frac{2G}{d(m_1+m_2)} }[/tex]
The relative velocity = v₁ + v₂ =[tex]v_1+v_2=m_1\sqrt{\frac{2G}{d(m_1+m_2)} } + m_2\sqrt{\frac{2G}{d(m_1+m_2)} } = (m_1+m_2)\sqrt{\frac{2G}{d(m_1+m_2)} }[/tex]
v₁ + v₂ = [tex](m_1+m_2)\sqrt{\frac{2G}{d(m_1+m_2)} }[/tex]
(b) The kinetic energy KE = [tex]\frac{1}{2}mv^2[/tex]
[tex]KE_1= \frac{1}{2} m_{1} v^2_1 \, \, \, KE_2= \frac{1}{2} m_{2} v^2_2[/tex]
Just before they collide, d = r₁ + r₂ = 3×10⁶+5×10⁶ = 8×10⁶ m
[tex]v_1 = 8\times10^{24}\sqrt{\frac{2\times6.67408 \times 10^{-11}} {8\times10^6(2.00\times10^{24}+8.00\times10^{24})} }[/tex] = 10333.696 m/s
[tex]v_2 = 2\times10^{24}\sqrt{\frac{2\times6.67408 \times 10^{-11}} {8\times10^6(2.00\times10^{24}+8.00\times10^{24})} }[/tex] =2583.424 m/s
KE₁ = 0.5×2.0×10²⁴× 10333.696² = 1.068×10³² J
KE₂ = 0.5×8.0×10²⁴× 2583.424² = 2.6696×10³¹ J.
