Answer:

Explanation:
1) Notation and info given
represent the density at the center of the planet
represent the densisty at the surface of the planet
r represent the radius
represent the radius of the Earth
2) Solution to the problem
So we can use a model to describe the density as function of the radius


So we can create a linear model in the for y=b+mx, where the intercept b=
and the slope would be given by 
So then our linear model would be

Since the goal for the problem is find the gravitational acceleration we need to begin finding the total mass of the planet, and for this we can use a finite element and spherical coordinates. The volume for the differential element would be
.
And the total mass would be given by the following integral

Replacing dV we have the following result:

We can solve the integrals one by one and the final result would be the following

Simplyfind this last expression we have:


![M=\pi r^3_{earth}[\rho_{surface}+\frac{1}{3}\rho_{center}]](https://tex.z-dn.net/?f=M%3D%5Cpi%20r%5E3_%7Bearth%7D%5B%5Crho_%7Bsurface%7D%2B%5Cfrac%7B1%7D%7B3%7D%5Crho_%7Bcenter%7D%5D)
And replacing the values we got:

And now that for any shape the gravitational acceleration is given by:

Answer:
A) The acceleration is zero
<em>B) The total distance is 112 m</em>
Explanation:
<u>Velocity vs Time Graph</u>
It shows the behavior of the velocity as time increases. If the velocity increases, then the acceleration is positive, if the velocity decreases, the acceleration is negative, and if the velocity is constant, then the acceleration is zero.
The graph shows a horizontal line between points A and B. It means the velocity didn't change in that interval. Thus the acceleration in that zone is zero.
A. To calculate the acceleration, we use the formula:

Let's pick the extremes of the region AB: (0,8) and (12,8). The acceleration is:

This confirms the previous conclusion.
B. The distance covered by the body can be calculated as the area behind the graph. Since the velocity behaves differently after t=12 s, we'll split the total area into a rectangle and a triangle.
Area of rectangle= base*height=12 s * 8 m/s = 96 m
Area of triangle= base*height/2 = 4 s * 8 m/s /2= 16 m
The total distance is: 96 m + 16 m = 112 m
Will this one work?...................
Answer:
The coupled velocity of both the blocks is 1.92 m/s.
Explanation:
Given that,
Mass of block A, 
Initial speed of block A, 
Mass of block B, 
Initial speed of block B, 
It is mentioned that if the two blocks couple together after collision. We need to find the common velocity immediately after collision. We know that due to coupling, it becomes the case of inelastic collision. Using the conservation of linear momentum. Let V is the coupled velocity of both the blocks. So,

So, the coupled velocity of both the blocks is 1.92 m/s. Hence, this is the required solution.