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A collision in orbit

Consider two nearly spherical Soyuz payload vehicles, in orbit about Earth, each with mass 9000 kg and diameter 4.0 m. They are initially at rest relative to each other, 10.0 m from center to center. (As we will see in Kepler’s Laws of Planetary Motion , both orbit Earth at the same speed and interact nearly the same as if they were isolated in deep space.) Determine the gravitational force between them and their initial acceleration. Estimate how long it takes for them to drift together, and how fast they are moving upon impact.

Strategy

We use Newton’s law of gravitation to determine the force between them and then use Newton’s second law to find the acceleration of each. For the estimate , we assume this acceleration is constant, and we use the constant-acceleration equations from Motion along a Straight Line to find the time and speed of the collision.

Solution

The magnitude of the force is

| F 12 | = F 12 = G m 1 m 2 r 2 = 6.67 × 10 −11 N · m 2 /kg 2 ( 9000 kg ) ( 9000 kg ) ( 10 m ) 2 = 5.4 × 10 −5 N.

The initial acceleration of each payload is

a = F m = 5.4 × 10 −5 N 9000 kg = 6.0 × 10 −9 m/s 2 .

The vehicles are 4.0 m in diameter, so the vehicles move from 10.0 m to 4.0 m apart, or a distance of 3.0 m each. A similar calculation to that above, for when the vehicles are 4.0 m apart, yields an acceleration of 3.8 × 10 −8 m/s 2 , and the average of these two values is 2.2 × 10 −8 m/s 2 . If we assume a constant acceleration of this value and they start from rest, then the vehicles collide with speed given by

v 2 = v 0 2 + 2 a ( x x 0 ) , where v 0 = 0 ,

so

v = 2 ( 2.2 × 10 −9 N ) ( 3.0 m ) = 3.6 × 10 −4 m/s.

We use v = v 0 + a t to find t = v / a = 1.7 × 10 4 s or about 4.6 hours.

Significance

These calculations—including the initial force—are only estimates, as the vehicles are probably not spherically symmetrical. But you can see that the force is incredibly small. Astronauts must tether themselves when doing work outside even the massive International Space Station (ISS), as in [link] , because the gravitational attraction cannot save them from even the smallest push away from the station.

A photo of an astronaut on a spacewalk is shown.
This photo shows Ed White tethered to the Space Shuttle during a spacewalk. (credit: NASA)

Check Your Understanding What happens to force and acceleration as the vehicles fall together? What will our estimate of the velocity at a collision higher or lower than the speed actually be? And finally, what would happen if the masses were not identical? Would the force on each be the same or different? How about their accelerations?

The force of gravity on each object increases with the square of the inverse distance as they fall together, and hence so does the acceleration. For example, if the distance is halved, the force and acceleration are quadrupled. Our average is accurate only for a linearly increasing acceleration, whereas the acceleration actually increases at a greater rate. So our calculated speed is too small. From Newton’s third law (action-reaction forces), the force of gravity between any two objects must be the same. But the accelerations will not be if they have different masses.

Got questions? Get instant answers now!

The effect of gravity between two objects with masses on the order of these space vehicles is indeed small. Yet, the effect of gravity on you from Earth is significant enough that a fall into Earth of only a few feet can be dangerous. We examine the force of gravity near Earth’s surface in the next section.

Questions & Answers

a tire 0.5m in radius rotate at constant rate 200rev/min. find speed and acceleration of small lodged in tread of tire.
Tahira Reply
hmm
Ishaq
100
Noor
define the terms as used in gravitational mortion 1:earth' satellites and write two example 2:parking orbit 3:gravitation potential 4:gravitation potential energy 5:escping velocity 6:gravitation field and gravitation field strength
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smooth or regular flow
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scalar field define with example
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the change in the position of an object in a particular direction is called displacement
Noor
The physical quantity which have both magnitude and direction are known as vector.
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Describe vector integral?
Malik
define line integral
Malik
Examples on how to solve terminal velocity
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ans:loading...
Lumai
the sideways pressure exerted by fluid is equal and canceled out.how and why?
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what is blackbody radiation
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Particles emitted by black holes
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what is oscillation
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Bernoulli's equation which applies a fluid flow states that P+h✓g+½PV²=k. where ✓=density. P=pressure. h=height. V= velocity. g=acceleration due to gravity. k= constant. Show that the equation is dimensionally constant and show the S.I unit for k.
amoni Reply
The nature of physics
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Physics deals with nature and its natural phenomenons
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physics explains more about mechanics and other natural phenomenon.
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Zahid Reply
Relative motion is the calculation of the motion of an object with regard to some other moving object. Thus, the motion is not calculated with reference to the earth, but is the velocity of the object in reference to the other moving object as if it were in a static state.
Lansana
I am unable to access the mcq can someone help me with it?
Harkamal Reply
What is free fall?
Barham Reply
when an object is falling under the the influence of the earth gravitational force, the term is called free fall.
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V=E½-P-½ where v; velocity, P; density and E; constant. Find dimension and it's units of E (constant)
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ML-3
LAWAL
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Triple temperature is the temperature at which melting ice and boiling water are at equilibrium
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Practice Key Terms 2

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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