1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Schach [20]
2 years ago
11

Two part?cles move about each other in circular orbits under the influence of gravita- tional forces, with a period t. Their mot

ion is suddenly stopped at a given instant of time, and they are then released and allowed to fall into each other. Prove that they collide after a time ?/4v2.
Physics
1 answer:
Pavlova-9 [17]2 years ago
3 0

It has been proven below that the two orbiting particles collided after a time τ/4√2.

<h3>How to prove the particles collided after a given time?</h3>

Assuming the particles to be point particles, the orbital period (time of fall) before the orbital motion is stopped for these particles would be derived by applying the Lagrangian equation for two orbiting particles:

L = T - V

L = 1/2MR² + 1/2μr² + Gm₁m₂/|r|     .....equation 1.

<u>Where:</u>

  • M = m₁ + m₂
  • μ = m₁m₂/m₁ + m₂

<u>Note:</u> The radius, r is constant in a circular orbit.

In Orbit Mechanics, the equation of relative motion is given by:

μr - μrθ = -Gm₁m₂/r²

Letting a = r, we have:

μaθ² = -Gm₁m₂/a²

Making θ the subject of formula and differentiating wrt t, we have:

\theta = a^{ \frac{3}{2} }[G(m_1 + m_2)]^{ \frac{1}{2} }\\\\\frac{d\theta}{dt}  = a^{ \frac{3}{2} }[G(m_1 + m_2)]^{ \frac{1}{2} }\\\\dt = \frac{a^{ \frac{3}{2} }}{[G(m_1 + m_2)]^{ \frac{1}{2} }} d\theta\\\\

Integrating over a full revolution, we have:

\int\limits^\tau_0  dt = \frac{a^{ \frac{3}{2} }}{[G(m_1 + m_2)]^{ \frac{1}{2} }} \int\limits^{2 \pi} _0d\theta\\\\\\\tau = \frac{2 \pi a^{ \frac{3}{2} }}{[G(m_1 + m_2)]^{ \frac{1}{2} }}.......equation 2.

Since the motion of the two orbiting particles is suddenly stopped (θ = 0) at a given instant of time, the equation of motion is then given by:

μr = -Gm₁m₂/r²

Multiplying both sides by 2r/μ, we would have:

2rr = -Gm₁m₂/μ × r/r²

In terms of dt, we would rewrite the equation as follows:

d/dt(r²) = -Gm₁m₂/μ × (dr/dt)/r²

Also, multiplying both sides by dt, we would have this integrated equation:

∫d/dt(r²)dt = -Gm₁m₂/μ × ∫(dr/dt)/r²dt

∫d(r²) = -Gm₁m₂/μ × ∫dr/r²

r² = 2G(m₁ + m₂)1/r + C

For the integration constant, we have:

C = -2G/a(m₁ + m₂).

So, r² = 2G(m₁ + m₂)(a - r)/ar

In terms of dt, we have:

dt=[\frac{2G}{a} (m_1+m_2)^\frac{-1}{2} ]\sqrt{\frac{r}{a-r} } dr\\\\T=\int\limits^T_0 dt=[\frac{2G}{a} (m_1+m_2)^\frac{-1}{2} ]\int\limits^0_a\sqrt{\frac{r}{a-r} } dr\\\\T =\int\limits^0_a\sqrt{\frac{r}{a-r} } dr

<u>Note:</u> Let the time for the two orbiting particles to collide be T.

By integrating the above through substitution method and substituting eqn. 2, we obtain:

T=\frac{1}{4\sqrt{2}  } \times  \frac{2 \pi a^{ \frac{3}{2} }}{[G(m_1 + m_2)] }}\\\\T=\frac{1}{4\sqrt{2}  } \times \tau\\\\T=\frac{\tau}{4\sqrt{2}  }

Time, T = τ/4√2 (proved).

Read more on orbital period here: brainly.com/question/13008452

#SPJ4

<u>Complete Question:</u>

Two particles move about each other in circular orbits under the influence of gravitational forces, with a period t. Their motion is suddenly stopped at a given instant of time, and they are then released and allowed to fall into each other. Prove that they collide after a time τ/4√2.

You might be interested in
Which of the waves has the smallest amplitude?
balandron [24]

Answer:

I think its the Blue wave, im not sure so dont take my word for it.

Explanation:

3 0
3 years ago
Who wrote the universal gravitation
ira [324]
I think it was Isaac Newton
3 0
3 years ago
An elevator was a full mass of 2400 kg has a power rating on 16.5 horsepower how much time will it take the elevator to lift a f
ehidna [41]
Jajjqjakwkwkwkjsjshdhdbxjixozkaknsns
6 0
3 years ago
If we heated the ball up and kept the ring room temperature, would the ball be able to fit through the ring? (1 point) Why or wh
Fed [463]

Answer:

no:

Explanation:

it would grow and no longer be able to fit through the loop due to the hot air expanding.

8 0
3 years ago
Define discrimination and write a sentence using the word.
seraphim [82]

Answer:

discrimination: prejudice towards a person/group based on their race, sex, age, and/or sexual orientation

Explanation:

People of color face discrimination because of the color of their skin.

5 0
3 years ago
Read 2 more answers
Other questions:
  • A force of 50 pounds is directed 30 from the horizontal, find the vector that represents this force.
    12·1 answer
  • What is the specific heat of silver if the temp ig a 15.4g sample of silver is increased frin 20 degrees celsius to 31.2 degrees
    5·1 answer
  • The precipitator uses a high voltage DC power supply to provide a set of wires with a positive charge and a set of plates with a
    11·1 answer
  • “Would you rather have toast or cereal for breakfast?” Lila's mother asks, allowing Lila to have a sense of control over her bre
    8·1 answer
  • Even at such low fractions of the speed of light these stars are moving quite quickly (thousands of kilometers a second) compare
    8·1 answer
  • Which of the following sequences correctly displays the energy transformation from the inside of a battery to the igniter inside
    12·1 answer
  • How fast must a 1000 kg car be moving to have a kinetic energy of 2.0×10^5j​
    5·1 answer
  • What cultular value does this melodic line portray?<br>a.jubilant<br>b.reserved<br>c.rich<br>d.poor​
    15·1 answer
  • Help it’s multiple choice 11 through 15 please!
    15·1 answer
  • What is the least dense planet in our solar system?.
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!