I have a few questions that I need help with please.

Show by algebraic reasoning that your gravitational acceleration toward an object of mass M a distance d away is a = GM/d2 and therefore doesn’t depend on your mass. (I don't understand this question)

What does it mean to say that angular momentum is conserved? Give an example you might see on Earth. How could conservation of angular momentum help astronomers understand the development of the solar system? (Could you please check the part of the question I answered and help me with the part that I didn't answer)

When angular motion is conserved it means that with no external torque, the product of rotational inertia and rotational velocity at one time will be the same as at any other time. An example your might see on Earth is the moon getting farther away.

Well, Newtons gravitation law is

force=GMm/d^2

but for you, mass m, your acceleration by falling in gravity must be

Force=ma or
a=F/m= GM/d^2

I don't know what the second question is, you made a statement.

Sure! I can help you with those questions.

1. To show algebraically that your gravitational acceleration toward an object of mass M a distance d away is given by a = GM/d^2, we can use Newton's law of universal gravitation. This law states that the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically, this can be expressed as:

F = G * (m1 * m2) / r^2

where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.

Now, let's consider an object with mass m experiencing the gravitational force from an object with mass M located at a distance d away. The force experienced by the object with mass m is given by F = G * (m * M) / d^2.

According to Newton's second law of motion, force (F) is equal to mass (m) multiplied by acceleration (a). So we can write:

F = m * a

Equating the expressions for the force, we have:

m * a = G * (m * M) / d^2

Simplifying the equation, we get:

a = G * M / d^2

Therefore, we have shown algebraically that the gravitational acceleration toward an object of mass M a distance d away is given by a = GM/d^2. And since mass m cancels out in the equation, it demonstrates that gravitational acceleration doesn't depend on your mass.

2. When we say that angular momentum is conserved, it means that the total angular momentum of a system remains constant if no external torques act on it. Angular momentum is a property that characterizes the rotational motion of an object and is defined as the product of its moment of inertia and its angular velocity.

An example you might see on Earth to understand conservation of angular momentum is a figure skater spinning on ice. When a figure skater performs a spin, they start with their arms extended and then pull them in closer to their body. As they bring their arms closer, their moment of inertia decreases. According to the conservation of angular momentum, the decrease in moment of inertia results in an increase in angular velocity, causing the skater to spin faster.

In the context of the development of the solar system, conservation of angular momentum is valuable to astronomers. As a cloud of gas and dust collapses under its own gravity to form a spinning protoplanetary disk, any small region that collapses will conserve its angular momentum. This conservation leads to the formation of a rotating disk, from which planets can eventually form. By studying the angular momentum of planets, scientists can gain insights into the processes that led to the formation and evolution of the solar system.

I hope this helps! Let me know if you have any further questions.