Given the two proportional relationships:
1. [tex]\( P \propto \frac{v^2}{x} \)[/tex]
2. [tex]\( x \propto v t \)[/tex]
Let us express [tex]\(P\)[/tex] in terms of [tex]\(v\)[/tex] and [tex]\(t\)[/tex].
### Step-by-Step Solution:
1. Express [tex]\( x \)[/tex] in terms of [tex]\( v \)[/tex] and [tex]\( t \)[/tex]:
Since [tex]\( x \propto v t \)[/tex], we can write:
[tex]\[
x = k_1 \cdot v \cdot t
\][/tex]
where [tex]\( k_1 \)[/tex] is a constant of proportionality.
2. Express [tex]\( P \)[/tex] in terms of [tex]\( v \)[/tex] and [tex]\( x \)[/tex]:
Given the relationship [tex]\( P \propto \frac{v^2}{x} \)[/tex], we can write:
[tex]\[
P = k_2 \cdot \frac{v^2}{x}
\][/tex]
where [tex]\( k_2 \)[/tex] is another constant of proportionality.
3. Substitute the expression for [tex]\( x \)[/tex] into the equation for [tex]\( P \)[/tex]:
Substitute [tex]\( x = k_1 \cdot v \cdot t \)[/tex] into [tex]\( P = k_2 \cdot \frac{v^2}{x} \)[/tex]:
[tex]\[
P = k_2 \cdot \frac{v^2}{k_1 \cdot v \cdot t}
\][/tex]
4. Simplify the expression:
[tex]\[
P = k_2 \cdot \frac{v^2}{k_1 \cdot v \cdot t} = \frac{k_2}{k_1} \cdot \frac{v^2}{v \cdot t} = \frac{k_2}{k_1} \cdot \frac{v}{t}
\][/tex]
5. Introduce a new constant [tex]\( k_3 \)[/tex]:
Let [tex]\( k_3 = \frac{k_2}{k_1} \)[/tex], where [tex]\( k_3 \)[/tex] is a new constant.
Therefore, we can write:
[tex]\[
P = k_3 \cdot \frac{v}{t}
\][/tex]
### Conclusion:
Thus, [tex]\( P \)[/tex] is directly proportional to [tex]\( \frac{v}{t} \)[/tex].
We have:
[tex]\[
P = k_3 \cdot \frac{v}{t}
\][/tex]
In this expression, [tex]\( k_3 \)[/tex] is the constant constant of proportionality. Given [tex]\( k_3 = 1 \)[/tex], it implies:
[tex]\[
P = \frac{v}{t}
\][/tex]
This is the detailed step-by-step solution for expressing [tex]\(P\)[/tex] in terms of [tex]\(v\)[/tex] and [tex]\(t\)[/tex].
