The variables x, v, and a have the dimensions of `[L],[L]//[T], and [L]//[T]^(2)`, respectively. These variables are related by an equation that has the form `v^(n)=2ax`, where n is an integer constant (1, 2, 3, etc.) without dimensions. What must be the value of n, so that both sides of the equation have the same dimensions? Explain your reasoning.

Options

A. 1
B. 3
C. 2
D. 4

Answer

To solve the problem, we need to ensure that both sides of the equation \( v^n = 2ax \) have the same dimensions. Let's break down the steps to find the value of \( n \).

### Step 1: Identify the dimensions of each variable
- The dimension of \( x \) is given as \( [L] \).
- The dimension of \( v \) (velocity) is \( [L][T]^{-1} \).
- The dimension of \( a \) (acceleration) is \( [L][T]^{-2} \).

### Step 2: Write down the dimensions of the left side of the equation
The left side of the equation is \( v^n \). Therefore, we need to find the dimensions of \( v^n \):
\[ \text{Dimensions of } v^n = ([L][T]^{-1})^n = [L^n][T]^{-n} \]

### Step 3: Write down the dimensions of the right side of the equation
The right side of the equation is \( 2ax \). Since 2 is a dimensionless constant, we can ignore it for dimensional analysis:
\[ \text{Dimensions of } 2ax = a \cdot x = ([L][T]^{-2}) \cdot [L] = [L^2][T]^{-2} \]

### Step 4: Set the dimensions of both sides equal to each other
Now we set the dimensions of both sides equal:
\[ [L^n][T]^{-n} = [L^2][T]^{-2} \]

### Step 5: Equate the powers of \( L \) and \( T \)
From the equation \( [L^n][T]^{-n} = [L^2][T]^{-2} \), we can equate the powers of \( L \) and \( T \):
1. For \( L \): \( n = 2 \)
2. For \( T \): \( -n = -2 \) (which also gives \( n = 2 \))

### Conclusion
Both conditions give us the same result, \( n = 2 \). Therefore, the value of \( n \) must be 2 for both sides of the equation to have the same dimensions.

### Final Answer
The value of \( n \) is \( 2 \).

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