Q25P - Fundamentals Of Physics

Question

During heavy rain, a section of a mountainside measuring 2.5 km horizontally, 0.80 km up along the slope, and 2.0 m deep slips into a valley in a mud slide. Assume that the mud ends up uniformly distributed over a surface area of the valley measuring 0.40 km × 0.40 km and that mud has a density of . What is the mass of the mud sitting above a 4.0 m² area of the valley floor?

Step-by-Step Solution

Answer

The mass of the mud sitting above the area of the valley floor is $1.9\times {10}^5 \text{kg}$

Step 1: Given data

Dimensions of section of mountainside are $2.5 \text{km}\times 0.80 \text{km}\times 2.0 \text{m}$

The dimensions of surface area of valley $0.40 \text{km}\times 0.40 \text{km}$

Density of mud ${\text{ρ=1900 kg/m}}^3$

The horizontal distance is A = 2.5 km = 2500 m , the height is,B = 0.80 km = 800m and the depth is, C = 2.0 m

Area of the valley floor is 

Step 2: Understanding the density of material

The density of a material is given by mass per unit volume. A volume is the amount of space occupied by an object in three-dimensional space.

The expression for density is given as: 
$\rho =\frac{m}{V}$… (i)

Here,  $\rho$ is the density, $m$ is the mass and $V$ is the volume

Step 3: Determination of the volume of the section

The volume of the section is calculated as: 

Thus, the volume of the section is $4\times {10}^6 {\text{m}}^3$

Step 4: Determination of the thickness of the mud

 

Let d be the thickness of the mud after it has uniformly distributed in the valley. 

So, the volume of the mud is,

$V=400 \text{m}\times 400 \text{m}\times d$

 

Since the mud is distributed uniformly, therefore, 

 $$\begin{gathered} 400 \text{m}\times 400 \text{m}\times d=4\times {10}^6 {\text{m}}^3 \\ d=25 \text{m} \end{gathered}$$

Step 5: To calculate mass of the part of mud

Calculate the volume of mud over area of $\text{4.0} {\text{m}}^2$

$$\begin{gathered} V_{\text{m}}=4.0 {\text{m}}^{\text{2}}\times d \\ =4.0 m^2\times 25 \text{m} \\ =100.0 {\text{m}}^{\text{3}} \end{gathered}$$

As each cubic meter corresponds to a mass of 1900 kg, the mass of that part of the mud can be calculated using the formula for the density.

Thus, the mass of that part of mud is $1.9\times {10}^5 \text{kg}$