Let's throw words having the syllable "-ject" into our collection of words, derived from "iacere" -- meaning "to throw." You've heard of "reject" and "eject"; here are several others tossed into the mix.
More Latin Love, Volume II lists: cadere, fluere, onym, and vertere!
ELA Common Core State Standard: "Use common, grade-appropriate Greek or Latin affixes and roots as clues to the meaning of a word."
of a mental act performed entirely within the mind
The connection between the prefix "sub-" (under)and "-ject," (throw) is indirect, a little hard to see. To look at something subjectively is to be under the influence of personal opinion, experience, self-serving motivation, or emotion that underlies judgment.
the goal intended to be attained (and which is believed to be attainable)
The word "objective" (literally: "that at which we are thrown") is used as a noun in the example sentence. As a noun, "objective" simply means "goal" and is used interchangeable with the words "goal," "target," or "aim." As an adjective, to be "objective" is to be able to look at something without any personal or emotional attachment, with just the facts thrown at you.
With the prefix "e-" meaning "out," and the root "ject-" meaning "to throw," it's clear to see that the word "eject" means "to throw out." However, when we remove the garbage from the house, we never say that we are "ejecting" it. The word "eject" is usually applied to either a forceful toss-out, or the removal of a member of the group, as in the example sentence.
With the prefix "in-" meaning "into," and the root "ject-" meaning "to throw," it's clear to see that the word "inject" means "to throw into," although this word is usually used in a medical sense, in its noun form, "injection," known colloquially as "a shot." Shots, are, after all, thrown in some way, hence the connection.
to believe especially on uncertain or tentative grounds
With the prefix "con-" meaning "with," and the root "ject-" meaning "to throw," with a little imagination, we can see how the word "conjecture" derives its meaning of "an informed guess." A conjecture lies somewhere between a hypothesis (an educated guess) and a wild guess: a conjecture is a guess based on some facts thrown together with a hunch or two.
Injured skier Lindsay Vonn flew home from Austria on Tiger Woods' private jet, only intensifying conjecture about their relationship.
—Seattle Times (Feb 16, 2013)
a weapon that is forcibly thrown or projected at a targets but is not self-propelled
With the prefix "pro-" meaning "forward" and the root "ject-" meaning "to throw," and the suffix "-ile" transforming a noun into a verb, it's clear to see how that word "projectile" means an object that is thrown, usually against a target. Bullets, cannon balls, and paper airplanes are examples of projectiles.
The dinosaurs were all-but certainly wiped out by a direct hit by a far smaller projectile 65 million years ago.
—Time (Feb 5, 2013)
the path followed by an object moving through space
With the prefix "tra-" being a form of the prefix "trans-" meaning "across," and the root "-ject" meaning "to throw," and the suffix "-ory" designating a noun, we can see how the word "trajectory" refers to a path of objects through space. But the word "trajectory" actually denotes more than that: A trajectory is a kind of pre-determined or predictable path, not only of physical objects, but of events in life as well. Key decisions that we make in life are said to set our trajectories.
With the prefix "de-" meaning, in this case, "down," and the root "ject-" meaning "to throw," we can see how the word "dejected" means "feeling low." To feel "dejected" is a little different from feeling "depressed." Usually, the word "dejected" is used when one's low spirits may be ascribed to a particular hurtful or disappointing event, while "depressed" usually refers to a more generalized and chronic sorrow.
With the prefix "ab-" meaning "away" and the root "ject-" meaning "to throw," we can see how the word "abject" means, truly, "thrown away." Usually used as an adjective to modify "poverty" ("abject poverty") the word "abject" connotes a hopeless state and is a sad word to even contemplate, especially when applied to large groups of people, which it usually is.