Question

Two spheres having masses Mand 2M and radii R and 3R, respectively, are simultaneously released from rest when the distance between their centers is 12R. Assume the two spheres interact only with each other and we wish to find the speeds with which they collide. (a) What two isolated system models are appropriate for this system? (b) Write an equation from one of the models and solve it for${\overrightarrow{\mathbf{v}}}_{\mathbf{1}}$ , the velocity of the sphere of mass M at any time after release in terms of $\overrightarrow{{\mathbf{v}}_{\mathbf{2}}}$, the velocity of 2M. (c) Write an equation from the other model and solve it for speed${\mathit{v}}_{\mathbf{1}}$ in terms of speed${\mathit{v}}_{\mathbf{2}}$ when the spheres collide. (d) Combine the two equations to find the two speeds ${\mathit{v}}_{\mathbf{1}}$and${\mathit{v}}_{\mathbf{2}}$ when the spheres collide.

Short answer

(a)The two appropriate isolated system models for the given system is the conservation of momentum model and conservation of energy model.

(b)$\overrightarrow{v_{1f}}=-2\overrightarrow{v_{2f}}$

(c)$v_{1f}=\sqrt{\frac{2GM}{3R}-2v_{2f}^2}$

(d) Speeds will be $v_{1f}=\frac{2}{3}\sqrt{\frac{GM}{R}}$and$v_{2f}=\frac{1}{3}\sqrt{\frac{GM}{R}}$ .

Step-by-step solution

Conservation of energy and conservation of momentum model 

Conservation of energy model states that in any isolated system remains constant. That means the initial kinetic K and potential U energy will be equal to the final kinetic and potential energy.

$K_i+U_i+=K_f+U_f$

Conservation of momentum model states that the momentum of any system will be constant until and unless any external force are getting applied on it.

${\left(m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2}\right)}_{initial}={\left(m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2}\right)}_{final}$

Here,

role="math" localid="1668153986745" $$\begin{gathered} {\text{m}}_{\text{1}}=\text{mass of sphere first} \\ \overrightarrow{{\text{v}}_{\text{1}}}=\text{velocity of sphere first} \\ {\text{m}}_{\text{2}}=\text{mass of sphere second} \\ \overrightarrow{{\text{v}}_{\text{2}}}=\text{velocity of sphere second} \end{gathered}$$

(a) Two appropriate isolated system models

The two appropriate isolated system models for the given system is the conservation of momentum model and conservation of energy model that will get applied on the system as this is consisting of two spheres.

(b) Equation In required form

By applying the conservation of the momentum model of the system we will get

$\begin{array}{rcl}{m}_1\overrightarrow{v_{1i}}+m_2\overrightarrow{v_{2i}}& =& m_1\overrightarrow{v_{1f}}+m_2\overrightarrow{v_{2f}}\\ 0+0& =& M\overrightarrow{v_{1f}}+2M\overrightarrow{v_{2f}}\\ \overrightarrow{v_{1f}}& =& -2\overrightarrow{v_{2f}}\\ & & \end{array}$

Suffix i = initial and suffix f = final

(c) Equation in required form

By applying the conservation of energy formula we will get

$\begin{array}{rcl}{K}_i+U_i+& =& K_f+U_f\\ 0-\frac{G{m}_1{m}_2}{r_i}& =& \frac{1}{2}{m}_1{v}_{1f}^2+\frac{1}{2}{m}_2{v}_{2f}^2-\frac{G{m}_1{m}_2}{r_f}\\ -\frac{GM\left(2M\right)}{12R}& =& \frac{1}{2}M{v}_{1f}^2+\frac{1}{2}2M{v}_{2f}^2-\frac{GM\left(2M\right)}{4R}\\ v_{1f}& =& \sqrt{\frac{2GM}{3R}-2v_{2f}^2}\end{array}$

Suffix i = initial and suffix f = final

(d) Speeds

By combining the solution part of the equation b and c we will get

$\begin{array}{rcl}2v_{2f}& =& \sqrt{\frac{2GM}{3R}-2v_{2f}^2}\\ 6v_{2f}^2& =& \frac{2GM}{3R}\\ v_{2f}& =& \frac{1}{3}\sqrt{\frac{GM}{R}}\\ & & \end{array}$

By putting the values,

$v_{1f}=\frac{2}{3}\sqrt{\frac{GM}{R}}$

These are the values for the speed.