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 ) (\boldsymbol{downarrow}) at (\boldsymbol{downarrow}) ( e_k ).

Explanation:

Step1: Analyze the function

The equation is \( v=\sqrt{\frac{2E_{k}}{m}} \), with \( m = 125 \) (a constant). So we can rewrite it as \( v=\sqrt{\frac{2}{125}E_{k}}=\sqrt{\frac{2}{125}}\cdot\sqrt{E_{k}} \). Let \( k=\sqrt{\frac{2}{125}} \) (a positive constant), then \( v = k\sqrt{E_{k}} \).

Step2: Determine the relationship

For a function of the form \( y = a\sqrt{x} \) where \( a>0 \), as \( x \) (here \( E_{k} \)) increases, \( y \) (here \( v \)) increases. Also, the rate of change: let's consider the derivative (or just analyze the shape). The function \( v(E_{k})=k\sqrt{E_{k}} \), the derivative \( v^\prime(E_{k})=\frac{k}{2\sqrt{E_{k}}} \), which decreases as \( E_{k} \) increases (since \( \sqrt{E_{k}} \) is in the denominator and increasing). So the rate at which \( v \) increases slows down as \( E_{k} \) increases, meaning \( v \) increases at a decreasing rate of \( E_{k} \).

Answer:

As \( E_{k} \) increases, \( v \) \(\boldsymbol{\text{increases}}\) at \(\boldsymbol{\text{a decreasing rate of}}\) \( E_{k} \).