# Object Function Homework

The two equations of motion reduces down to one equation of motion by considering the separation $x=x_2-x_1$ and the separating acceleration $\ddot{x} = \ddot{x}_2 -\ddot{x}_1$

$$\ddot{x} = -\frac{G (m_1+m_2)}{x^2}$$

or $\ddot{x} = -K/x^2$ with $K=G (m_1+m_2)$

This can be re-written as $\frac{{\rm d} \dot{x}}{{\rm d} t} =\frac{{\rm d} \dot{x}}{{\rm d} x} \frac{{\rm d} x}{{\rm d} t} =\frac{{\rm d} \dot{x}}{{\rm d} x} \dot x = -\frac{K}{x^2}$

$$\int \dot{x} {\rm d} \dot{x} = -\int \tfrac{K}{x^2}\,{\rm d}x + C_1$$ $$\frac{1}{2} \dot{x}^2 = C_1 + \frac{K}{x}$$

If initially the bodies are at rest, separated by $d$ then

$$\frac{1}{2} \dot{x}^2 = -\frac{K}{d} + \frac{K}{x}$$ or $$\dot{x} = \sqrt{\frac{2 K (d-x)}{d\, x}}$$

This has a solution for time $t$ as a function of separation $x$ of

$$t= \sqrt{ \frac{d^3}{2 K}} \cos^{-1} \left( \sqrt{\frac{x}{d}}\right) - \sqrt{ \frac{d^2 x-d x^2}{2 K}}$$

This means the time to reach collision $x=0$ is

$$t_C = \frac{\pi}{2}\sqrt{ \frac{d^3}{2 K}} = \frac{\pi}{2}\sqrt{ \frac{d^3}{2 G (m_1+m_2)}}$$

answered Nov 10 '15 at 6:41

## What is an Object Pronoun?

An object pronoun is a type of personal pronoun that is normally used as a grammatical object, either as the direct or indirect object of a verb, or as the object of a preposition. These pronouns always take the objective case, whether they are indirect object pronouns or direct object pronouns.

The seven basic pronouns take on different forms when used as object pronouns rather than as subject pronouns:

• I becomes Me
• You becomes You (tnis rule applies for singular and plural use)
• He becomes Him
• She becomes Her
• It becomes It
• We becomes Us
• They becomes Them

If you know how to find the object of a sentence, then you will find it very easy to identify an object pronoun when you see one. To find the object of a sentence, locate the nouns and verbs and mentally separate them from the rest of the words the sentence contains. Both the subject and object of the sentence will be nouns or pronouns. The subject will be the one doing the action the verb describes. The object is the noun or pronoun receiving the action. Making things even simpler, when the object is not a noun, it’s an object pronoun.

Just like subject pronouns, object pronouns can be singular or plural, masculine, feminine, or gender neutral. The masculine or feminine subject pronoun is used whenever the gender is known. When referring to the weather, temperature, time, an inanimate object or a child or animal of undetermined gender, the neuter form “it” can be used.

## Object Pronoun Examples

In the following examples, the subject is in bold, the verb is in italics, and the object pronoun is in bold and is underlined.

1. Bob took her to work Monday.
2. Will you please tell them to come in?
3. He told you a lie about where he was Saturday.
4. Our grandparents gaveus candy and our teeth are just fine.

## Object Pronouns Exercises

1. Him
2. Her
3. You
4. Them
2. The plate shattered when John dropped __________ on the floor.
1. Him
2. Her
3. Them
4. It
3. Be careful; he lied to __________ before and he may do it again.
1. Us
2. It
3. We
4. They
4. Where are Jill and Cherie? Didn’t you invite __________?
1. Us
2. Them
3. They
4. Her
5. The spider bit __________ on my ankle.
1. You
2. Me
3. Her
4. It
6. I heard that Jeremy was cut from the team just because Tyler doesn’t like __________.
1. Him
2. Her
3. It
4. Them
7. The bread is stale. You can feed __________ to the birds.
1. Them
2. It
3. Him
4. Her
8. The movie was hilarious! We really liked __________.
1. Him
2. Them
3. Her
4. It