An athlletic coach told his team that muscle times speed equals power. What dimesions does he view for muscle?

Options

A. `MLT^(-2)`
B. `ML^(2)T^(-2)`
C. `MLT^(-2)`
D. L

Explanation

To solve the problem, we need to find the dimensions of "muscle" based on the relationship given by the athletic coach: muscle times speed equals power. Let's break this down step by step.

### Step 1: Understand the relationship
The coach states that:
\[ \text{Muscle} \times \text{Speed} = \text{Power} \]

Let’s denote the dimension of muscle as \( [M] \).

### Step 2: Write the dimensions of speed
Speed is defined as distance traveled per unit time. The dimension of speed can be expressed as:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{L}{T} \]
Thus, the dimension of speed is:
\[ [\text{Speed}] = [L][T^{-1}] \]

### Step 3: Write the dimensions of power
Power is defined as the rate of doing work or energy transfer per unit time. The dimension of power can be expressed as:
\[ \text{Power} = \frac{\text{Work}}{\text{Time}} \]
The dimension of work (or energy) is:
\[ \text{Work} = \text{Force} \times \text{Distance} = [M][L][T^{-2}][L] = [M][L^2][T^{-2}] \]
Thus, the dimension of power is:
\[ [\text{Power}] = \frac{[M][L^2][T^{-2}]}{[T]} = [M][L^2][T^{-3}] \]

### Step 4: Set up the equation with dimensions
From the relationship given:
\[ [M] \times [\text{Speed}] = [\text{Power}] \]
Substituting the dimensions we have:
\[ [M] \times [L][T^{-1}] = [M][L^2][T^{-3}] \]

### Step 5: Solve for the dimension of muscle
Now we can express this as:
\[ [M] \times [L][T^{-1}] = [M][L^2][T^{-3}] \]
To isolate the dimension of muscle, we can rearrange the equation:
\[ [M] = \frac{[M][L^2][T^{-3}]}{[L][T^{-1}]} \]

### Step 6: Simplify the right side
\[ [M] = [M] \times \frac{[L^2]}{[L]} \times \frac{[T^{-3}]}{[T^{-1}]} \]
This simplifies to:
\[ [M] = [M] \times [L^{2-1}][T^{-3+1}] = [M] \times [L^1][T^{-2}] \]

### Step 7: Cancel out the common dimension
Since we have \( [M] \) on both sides, we can cancel it out (assuming \( [M] \neq 0 \)):
\[ 1 = [L^1][T^{-2}] \]
Thus, the dimension of muscle can be expressed as:
\[ [M] = [L^1][T^{-2}] \]

### Final Answer
Thus, the dimension of muscle is:
\[ [M][L^1][T^{-2}] \]