Which of the following is dimensionally incorrect?

Options

A. u = v - at
B. `s-ut=1/2at^(2)`
C. `u^(2)=2a(g t-1)`
D. `v^(2)-u^(2)=2as`

Answer

To determine which of the given equations is dimensionally incorrect, we will analyze each option step by step.

### Step 1: Analyze the first equation
**Equation:** \( U = V - 80 \)

- **LHS (Left-Hand Side):**
- \( U \) is a velocity, so its dimension is \( [U] = [L][T^{-1}] \) (length per time).

- **RHS (Right-Hand Side):**
- \( V \) is also a velocity, so its dimension is \( [V] = [L][T^{-1}] \).
- The term \( 80 \) must have the same dimension as \( V \) to be subtracted from it. However, \( 80 \) is a numerical constant without any units, which makes it dimensionless.

- **Conclusion for Step 1:**
- The LHS has dimension \( [L][T^{-1}] \) while the RHS has dimension \( [L][T^{-1}] - \text{(dimensionless)} \), which is incorrect. Therefore, this equation is dimensionally incorrect.

### Step 2: Analyze the second equation
**Equation:** \( U^2 = 2A \cdot GT - 1 \)

- **LHS:**
- \( U^2 \) is the square of velocity, so its dimension is \( [U^2] = [L^2][T^{-2}] \).

- **RHS:**
- \( 2A \) is a constant multiplied by acceleration \( A \). The dimension of acceleration is \( [A] = [L][T^{-2}] \).
- \( G \) is also an acceleration (gravitational), so its dimension is \( [G] = [L][T^{-2}] \).
- \( T \) is time, so its dimension is \( [T] = [T] \).

Combining these:
- The dimension of \( 2A \cdot GT \) is:
\[ [2A \cdot GT] = [L][T^{-2}] \cdot [L][T^{-2}] \cdot [T] = [L^2][T^{-3}] \]

- The term \( -1 \) is dimensionless and cannot be subtracted from a term with dimensions.

- **Conclusion for Step 2:**
- The LHS has dimension \( [L^2][T^{-2}] \) while the RHS has dimension \( [L^2][T^{-3}] - \text{(dimensionless)} \), which is incorrect. Therefore, this equation is dimensionally incorrect.

### Step 3: Analyze the third equation
**Equation:** \( b^2 - u^2 = 2as \)

- **LHS:**
- \( b^2 \) and \( u^2 \) are both squares of velocities, so their dimensions are \( [L^2][T^{-2}] \).

- **RHS:**
- \( 2a \) is a constant multiplied by acceleration \( a \), which has dimension \( [L][T^{-2}] \).
- \( s \) is displacement, which has dimension \( [L] \).

Combining these:
- The dimension of \( 2as \) is:
\[ [2as] = [L][T^{-2}] \cdot [L] = [L^2][T^{-2}] \]

- **Conclusion for Step 3:**
- Both LHS and RHS have the same dimension \( [L^2][T^{-2}] \), so this equation is dimensionally correct.

### Step 4: Analyze the fourth equation
**Equation:** \( b^2 - u^2 = 2as \)

- This is the same as the previous equation, and we already determined it to be dimensionally correct.

### Final Conclusion
The dimensionally incorrect equations are:
1. \( U = V - 80 \) (1st equation)
2. \( U^2 = 2A \cdot GT - 1 \) (2nd equation)