When viewed from the non-inertial frame of one body only the trajectory of the opposite body is apparent ; Keplerian elements describe these non-inertial trajectories.

Translating this analogy into space, then, a " stationary " observer sees a planet moving left at speed U and a spaceship moving right at speed v. If the spaceship has the proper trajectory, it will pass close to the planet, moving at speed U + v relative to the planet's surface because the planet is moving in the opposite direction at speed U. When the spaceship leaves orbit, it is still moving at U + v relative to the planet's surface but in the opposite direction ( to the left ).

When it comes to the exact pairwise collision detection, this is highly trajectory dependent, and one almost has to use a numerical root-finding algorithm to compute the instant of impact.

When one particle scatters off of another, altering its trajectory, there are two ways to think about the process.

When it first appeared, the Lebel's 8 × 50R smokeless ammunition allied to its longer range and flatter trajectory brought a revolution in infantry armament.

When the bomb was delivered by low-level toss bombing, the aircraft was almost always at a lower altitude than the burst height ; so in effect, the bomb was not really " dropped ", but was released and " flew " upwards in a ballistic trajectory, to detonate when it reached the required altitude.

When an object bobs up and down on a ripple in a pond, it experiences an elliptical trajectory.

When satisfied along a trajectory, Pontryagin's minimum principle is a necessary condition for an optimum.

When development is disturbed by external forces, the embryo attempts to regulate its growth and differentiation by returning to its normal developmental trajectory.

When a golf ball is hit, the impact, which lasts less than a millisecond, determines the ball ’ s velocity, launch angle and spin rate, all of which influence its trajectory ( and its behavior when it hits the ground ).

When the object approaches Earth closely at either end of its trajectory, its apparent direction changes.

When no flags are available, a small leaf or other small light object can be dropped from shoulder height, and the object is then pointed at by the shooter ; the angle between his arm and his torso can provide an equivalent wind speed estimation as a wind flag, although it will not be at the mid-range location along the bullet's trajectory.

When it is close enough for some of the reflected laser energy from the target to reach it, a laser seeker detects which direction this energy is coming from and adjusts the projectile trajectory towards the source.

When we predict where a ball is going to land based on its current trajectory, we are taking the physical stance.

When the missile is in this transition state, the first stage booster sheds off, and the missile automatically adapts its " free " trajectory and further reaches its maximum altitude ( without a constant velocity ); the dominating force here is just the earth's gravity.

When the curvature of the Earth is taken into account it is found that, on average, radio waves whose initial trajectory is towards the optical horizon will in fact follow a path which does not return to the Earth's surface at the horizon, but slightly beyond it.

When the shield is struck, it returns the attack as a concentrated explosion that is automatically returned along the attack's opposing trajectory.

If the relative priorities of these substituents need to be established, R takes priority over S. When this happens, the descriptor of the stereocenter is a lowercase letter ( r or s ) instead of the uppercase letter normally used.

When r < sub > cov </ sub > is expressed in angstroms,

When r is small, N ( r ) is large.

When integrated over the radius, this implies that for b = 1, the total number of stars is proportional to r < sup > 2 </ sup >.

When proper units are used for tangential speed v, rotational speed ω, and radial distance r, the direct proportion of v to both r and ω becomes the exact equation:

When ∆ r is small compared to R, the terms after the first residual term are very small and can be neglected, giving the approximate tidal acceleration ( axial ) for the distances ∆ r considered, along the axis joining the centers of m and M:

When calculated in this way for the case where ∆ r is a distance along the axis joining the centers of m and M, is directed outwards from to the center of m ( where ∆ r is zero ).

When the actual transmitted ( incident ) power and the reflected power are known ( i. e., through measurements and / or calculations ), then the return loss in dB can be calculated as the difference between the incident power P < sub > i </ sub > ( in dBm ) and the reflected power P < sub > r </ sub > ( in dBm ),

When, despite the treaty, the Xiongnu continued to raid Han borders, Emperor Wu of Han ( r. 141 – 87 BCE ) launched several military campaigns against them.

When a request by Kushan ruler Vima Kadphises ( r. c. 90 – c.

When Empress Dowager Deng died, Emperor An ( r. 106 – 125 CE ) was convinced by the accusations of the eunuchs Li Run ( 李閏 ) and Jiang Jing ( 江京 ) that Deng and her family had planned to depose him.

where C is an arbitrary path connecting the point with zero potential to r. When the curl is zero, the line integral above does not depend on the specific path C chosen but only on its endpoints.

When the same roots occur multiple times, the terms in this formula corresponding to the second and later occurrences of the same root are multiplied by increasing powers of n. For instance, if the characteristic polynomial can be factored as ( x − r )< sup > 3 </ sup >, with the same root r occurring three times, then the solution would take the form

When does it have a solution of the same general form as a < sub > n </ sub > = r < sup > n </ sub >?

When a tube of a narrow bore, often called a capillary tube, is dipped into a liquid and the liquid wets the tube ( with zero contact angle ), the liquid surface inside the tube forms a concave meniscus, which is a virtually spherical surface having the same radius, r, as the inside of the tube.

When Xuanzong's son Li Heng ascended the throne as Emperor Suzong of Tang ( r. 756 – 762 ) commanderies once more became prefectures.

* When a long vowel is followed by " r ", some speakers of Indian English usually use a monophthong, instead of the diphthong used for many such words in many other accents.

When changing to a different coordinate system ( for example Lemaitre coordinates, Eddington-Finkelstein coordinates, Kruskal-Szekeres coordinates, Novikov coordinates, or Gullstrand – Painlevé coordinates ) the metric becomes regular at r = r < sub > s </ sub > and can extend the external patch to values of r smaller than r < sub > s </ sub >.

When there are only two means to compare, the t-test and the ANOVA F-test are equivalent ; the relation between ANOVA and t is given by F = t < sup > 2 </ sup >.

When the evolution map Φ < sup > t </ sup > ( or the vector field it is derived from ) depends on a parameter μ, the structure of the phase space will also depend on this parameter.

When his eighty-year-old son died, the father mourned: “ I always told him he wouldn ’ t live long, poor boy.

When the domain of the input function is time ( t ), and the domain of the output function is ordinary frequency, the transform of function s ( t ) at frequency ƒ is given by the complex number:

When s < sub > P </ sub >( t ), is expressed as a periodic summation of another function, s ( t ):

When a friend criticized him for denigrating Weiss, a man with an exemplary military record, " he explained cynically that he wasn ’ t in the least interested in Weiss, only in the propaganda effect.

When evaluating the integral, t is held constant, and so it is considered a parameter.

When asked, “ Don ’ t all religions teach the same thing?

When Gestapo agents were at his door he turned to his wife, who was crying, and said " Don ´ t cry, we were living in fear, but from now on we will live in hope ".

As Sørensen stated, “ When you meet others who have gone out on their own, it doesn ’ t seem that crazy .”

When asked if he would be re-recording some of Dan Nelson ’ s vocals he stated " That is what they want to do, and it ’ s been a conflict for me, to be quite frank, simply because I kind of felt like I wasn ’ t part of the creation of it.

When Linda Evangelista mentioned to Vogue that " we don ’ t wake up for less than $ 10, 000 a day ", she may have been playfully pretending the role of an up-scale union representative, but the 1990 comment became the most notorious quote in modeling history.

* The ' I Didn ’ t Know The Way To Kings Cross When I First Came Here But Look At Me Now ' Book By William Rushton, Author, Artist And Beer-Drinker Extrodinary William Rushton ( New England Library, 1966 )

When an object rotates about its axis, the motion cannot simply be analyzed as a particle, since in circular motion it undergoes a changing velocity and acceleration at any time ( t ).

When Agrippa reached the commandment of that “ you may not put a foreigner over you ” as king, his eyes ran with tears, but they said to him, “ Don ’ t fear, Agrippa, you are our brother, you are our brother !” The king would read from up through the shema (), and then the portion regarding tithes (), the portion of the king (), and the blessings and curses ().

When Clytemnestra sees Achilles she brings up the marriage, however Achilles doesn ’ t know what she is talking about and slowly the truth comes out about Agamemnon ’ s true plan.

They thus sang his fate: Clotho said that he would be noble, Lachesis that he would be brave, but Atropos looking at a brand burning on the hearth and said, “ He will live only as long as this brand remains unconsumed .” When Althaea, the mother, heard this, she leaped from the bed, put out the fatal brand, and buried it in the midst of the palace, so that it shouldn ’ t be destroyed by fire.

When asked what the mood of the people in Pineville was concerning the school consolidation, Alumbaugh said, “ I don ’ t remember it being a great deal ; most people considered it inevitable it was going to happen sooner or later.

When blowing the sh ' varim-teruah, the t ' qiah is lengthened to be slightly longer the duration of the sh ' varim and the teruah together.

When he heard the music he couldn ’ t dance to it, but he just started to shake all over.