Question

consider the equation $v = \sqrt{\frac{2e_k}{m}}$, where $m = 125$. complete the statement below about the relationship between $e_k$ and $v$. as $e_k$ increases, $v$ dropdown with options increases, decreases at dropdown $e_k$.

Explanation:

Step1: Analyze the formula structure

The formula is \( v = \sqrt{\frac{2E_k}{m}} \), and \( m = 125 \) (a constant). So the formula simplifies to \( v=\sqrt{\frac{2}{125}E_k} \), which can be written as \( v = \sqrt{\frac{2}{125}}\cdot\sqrt{E_k} \). The coefficient \( \sqrt{\frac{2}{125}} \) is a positive constant.

Step2: Determine the relationship type

In a function of the form \( y = k\sqrt{x} \) where \( k>0 \), as \( x \) (the independent variable) increases, \( y \) (the dependent variable) also increases. Here, \( v \) is like \( y \), \( E_k \) is like \( x \), and \( \sqrt{\frac{2}{125}} \) is the positive constant \( k \). So as \( E_k \) (the input) increases, \( v \) (the output) increases, and the relationship is proportional (specifically, a square - root proportionality, but the key is the increasing nature with \( E_k \) as the variable).

Answer:

First blank: increases; Second blank: a rate proportional to the square root of (or related to the square root of, but the main idea is that \( v \) increases with \( E_k \) in a way related to the square root of \( E_k \), and the first part is "increases" as the primary relationship direction)