Question

We watch two identical astronomical bodies Aand B, each of mass m, fall toward each other from rest because of the gravitational force on each from the other. Their initial center-to-center separation isR_i. Assume that we are in an inertial reference frame that is stationary with respect to the center of mass of this two body system. Use the principle of conservation of mechanical energy (K_f+ U_f=K_i +U_i ) to find the following when the center-to-center separation is 0.5Ri:

(a) the total kinetic energy of the system,

(b) the kinetic energy of each body,

(c) the speed of each body relative to us, and

(d) the speed of body Brelative to body A. Next assume that we are in a reference frame attached to body A(we ride on the body). Now we see body Bfall from rest toward us. From this reference frame, again use$\mathit{K}\mathit{f}\mathbf{}\mathbf{+}\mathbf{}\mathit{U}\mathit{f}\mathbf{}\mathbf{=}\mathbf{}\mathit{K}\mathit{i}\mathbf{}\mathbf{+}\mathit{U}\mathit{i}$to find the following when the center-to-center separation is$\mathbf{0}\mathbf{.}\mathbf{5}\mathit{R}\mathit{i}$:

(e) the kinetic energy of body Band

(f) the speed of body Brelative to body A.

(g) Why are the answers to (d) and (f) different? Which answer is correct?

Short answer

a)The total kinetic energy of the system is $K_{total}=\frac{G{m}^2}{R_i}$

b) The kinetic energy of each body is $KE=\frac{G{m}^2}{2R_i}$

c)The speed of each body relative to the inertial frame of reference is$V=\sqrt{\frac{Gm}{R_i}}$

d)The speed of body B relative to body A is $V_{BA}=2\sqrt{\frac{Gm}{R_i}}$

e) Kinetic energy of body B, when the frame of reference is body A, is$K_B=\frac{G{m}^2}{R_i}$

f) The speed of body B relative to body A, measured from frame of reference of A, is$V_B=\sqrt{\frac{2Gm}{R_i}}$

g) The answers of part d and f are different because the frames of referenceinthe twopartsaredifferent. The answer of part d is correct.

Step-by-step solution

Listing the given quantities

The mass of each body is m

The initial separation distance is R_i

The final separation distance is 0.5 R_i.

Understanding the concept of conservation of energy

Using the law of conservation of energy, we can find the total K.E of the system. From this, we can find the kinetic energy and velocity of each body relative to the inertial frame of reference or relative to the other body.

Formulae:

$$\begin{gathered} K_i+U_i=K_f+U_f \\ K=\frac{1}{2}m{v}^2 \\ U=-\frac{GMm}{R} \end{gathered}$$

(a)Calculations of totalkinetic energy of system

We have the equation for conservation of energy as

$$\begin{gathered} K_i+U_i=K_f+U_f \\ 0-\frac{G{m}^2}{R_i}=K_{total}-\frac{G{m}^2}{0.5R_i} \\ {K}_{total}=\frac{G{m}^2}{R_i} \end{gathered}$$

The total kinetic energy of the system is $K_{total}=\frac{G{m}^2}{R_i}$.

(b)Calculations of kinetic energy of each body

As the mass of each body is the same, their speed should also be the same.

So, the kinetic energy of each body is

$$\begin{gathered} KE=\frac{1}{2}{K}_{total} \\ KE=\frac{G{m}^2}{2R_i} \end{gathered}$$

The kinetic energy of each body is $KE=\frac{G{m}^2}{2R_i}$.

(c)Calculations of the speed of each body relative to the inertial frame of reference

We have the equation for KE as.$K=\frac{1}{2}m{v}^2$

By rearranging this equation for velocity and using the kinetic energy equation from part b, we get

$v=\sqrt{\frac{Gm}{R_i}}$

The speed of each body relative to the inertial frame of reference is $v=\sqrt{\frac{Gm}{R_i}}$.

(d)Calculations of the speed of body B relative to body A

The relative speed of B with respect to A, is $v_{BA}=2v$.

$v_{BA}=2\sqrt{\frac{Gm}{R_i}}$

The speed of body B relative to body A is $v_{BA}=2\sqrt{\frac{Gm}{R_I}}$.

(e)Calculations of the kinetic energy of body B when the frame of reference isbody A

In this part, the frame of reference is body A, so the total kinetic energy will be equal to the kinetic energy of body B.

Hence

$$\begin{gathered} K_B=K \\ {K}_B=\frac{D{m}^2}{R_i} \end{gathered}$$

Kinetic energy of body B when the frame of reference isbody A is$K_B=\frac{D{m}^2}{R_i}$.

(f)Calculations of the speed of body B relative to body A

In a similar way to part d,we have

$k_B=\frac{1}{2}m{v^2}_B$

Solving

$v_B=\sqrt{\frac{2Gm}{R_i}}$

The speed of body B relative to body A, measured from frame of reference of A, is$V_B=\sqrt{\frac{2Gm}{R_i}}$

(g)Explanation

When weconsider bodyA a frame of reference, we neglect the accelerated motion of body A. So, the answer of part f is incorrect, and the answer ofpart d is correct.