Short answer and validation with program

1

The essence of the game object movement is to change the coordinates of the game object itself.
In game development, we can transform game objects by matrix transformation, such as:

public class TranslationScript : MonoBehaviour
{
	// The migration direction vector of the game object, whose modulus length is the translation speed, and the vector direction is the translation direction.
	Vector3 translation = new Vector3(1, 0, 0);

	void Update()
	{
		// Each frame moves the game object, and the moving distance needs to consider the frame length.
		transform.Translate(Time.deltaTime * translation);
	}
}

public class RotationScript : MonoBehaviour
{
	// The rotation vector of the game object is determined by the length of the vector module. The game object will rotate counterclockwise along the axis of the straight line in the direction of the vector.
	Vector3 translation = new Vector3(1, 0, 0);

	void Update()
	{
		// For each frame of moving game object, the rotation angle should consider the frame length.
		transform.Translate(Time.deltaTime * translation);
	}
}

2

Method 1

It is implemented by Translate method.
In the two-dimensional coordinate system, the parabolic motion equation of the object is:
x=vxty=vyt+12at2 \begin{aligned} x&=v_xt\\ y&=v_yt+\frac{1}{2}at^2 \end{aligned} xy​=vx​t=vy​t+21​at2​
Then the motion vector of each time interval t Delta t t is:
x2−x1=vxt2−vxt1=vxΔty2−y1=vyt2+12at22−vyt1+12at12=vyΔt+12a(Δt2+2t1Δt) \begin{aligned} x_2-x_1&=v_xt_2-v_xt_1=v_x\Delta t\\ y_2-y_1&=v_yt_2+\frac{1}{2}at_2^2-v_yt_1+\frac{1}{2}at_1^2=v_y\Delta t+\frac{1}{2}a(\Delta t^2+2t_1\Delta t) \end{aligned} x2​−x1​y2​−y1​​=vx​t2​−vx​t1​=vx​Δt=vy​t2​+21​at22​−vy​t1​+21​at12​=vy​Δt+21​a(Δt2+2t1​Δt)​
Then we just need to pass code validation:

public class ProjectileMotionScript : MonoBehaviour
{
	// Initial velocity
    public Vector3 v; // Initial speed can be set to (10, 0, 0) in Unity interface.
    public float g; // Gravity acceleration, which can be set to 9.8 at Unity interface
    private float t = 0;
    
    void Update()
    {
        transform.Translate(v * Time.deltaTime);
        transform.Translate(0, -0.5f * g * (Time.deltaTime * Time.deltaTime + 2 * Time.deltaTime * t), 0);
        t += Time.deltaTime;
    }
}

3

We can add Trail Renderer components to each planet to track its trajectory.

1. Firstly, we add Rotate AroundScript to each planet. As shown below, the game object will rotate around its parent object. We can set the rotation plane and the rotation rate in Unity panel by setting direction and speed.

public class RotateAroundScript : MonoBehaviour
{
    public Vector3 direction;
    public float speed;

    void Update()
    {
        transform.RotateAround(transform.parent.position, direction * Time.deltaTime, speed * Time.deltaTime);
    }
}

1. Then we add the RotationScript component for each planet, as shown below, to allow direction and speed to be set through the Unity panel to control the rotation angle and rotation rate.

public class RotationScript : MonoBehaviour
{
    public Vector3 direction;
    public float speed;

    // Update is called once per frame
    void Update()
    {
        transform.Rotate(direction * Time.deltaTime, speed * Time.deltaTime);
    }
}

Programming practice

See https://github.com/huanghongxun/3D-Programming-And-Design/tree/master/homework2/PriestsAndDevils.

Thinking questions

// Let t rotate with specific axis and angle.
void Rotate(Transform t, Vector3 axis, float angle)
{
	var rot = Quaternion.AngleAxis(angle, axis);
    t.position = rot * t.position;
    t.rotation *= rot;
}

// Let t rotate around center with specific axis and angle.
void RotateAround(Transform t, Vector3 center, Vector3 axis, float angle)
{
	var position = t.position;
    var rot = Quaternion.AngleAxis(angle, axis);
    var direction = position - center;
    direction = rot * direction;
    t.position = center + direction;
    t.rotation *= rot;
}