Answered

Given that [tex]P \propto \frac{v^2}{x}[/tex] and [tex]x \propto v t[/tex], express [tex]P[/tex] in terms of [tex]v[/tex] and [tex]t[/tex].



Answer :

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].