For a body in a uniformly accelerated motion, the distance of the body from a reference point at time 't' is given by `x = at + bt^2 + c`, where a, b , c are constants. The dimensions of 'c' are the same as those of
`(A) x " " (B) at " " (C ) bt^2 " " (D) a^2//b`

Options

A. A
B. A & B
C. A, B & C
D. A, B, C & D

Answer

To solve the problem, we need to analyze the equation given for the distance of a body in uniformly accelerated motion:

\[ x = at + bt^2 + c \]

where \( a \), \( b \), and \( c \) are constants.

### Step 1: Identify the dimensions of each term in the equation.

1. **Dimension of \( x \)**:
- \( x \) represents distance, so its dimension is \([L]\) (length).

2. **Dimension of \( at \)**:
- \( a \) is a constant multiplied by time \( t \).
- The dimension of time \( t \) is \([T]\).
- Thus, the dimension of \( at \) is \([L] = [a][T]\).
- Therefore, the dimension of \( a \) must be \([L][T^{-1}]\).

3. **Dimension of \( bt^2 \)**:
- \( b \) is a constant multiplied by \( t^2 \).
- The dimension of \( t^2 \) is \([T^2]\).
- Thus, the dimension of \( bt^2 \) is \([L] = [b][T^2]\).
- Therefore, the dimension of \( b \) must be \([L][T^{-2}]\).

4. **Dimension of \( c \)**:
- Since \( c \) is added to \( at \) and \( bt^2 \), it must have the same dimension as these terms.
- Therefore, the dimension of \( c \) is also \([L]\).

### Step 2: Compare the dimensions of \( c \) with the options provided.

Now we will compare the dimension of \( c \) with the dimensions of the options given:

- **Option A: Dimension of \( x \)**:
- Dimension of \( x \) is \([L]\).

- **Option B: Dimension of \( at \)**:
- Dimension of \( at \) is \([L]\).

- **Option C: Dimension of \( bt^2 \)**:
- Dimension of \( bt^2 \) is \([L]\).

- **Option D: Dimension of \( \frac{a^2}{b} \)**:
- We know:
- Dimension of \( a \) is \([L][T^{-1}]\).
- Dimension of \( b \) is \([L][T^{-2}]\).
- Therefore, dimension of \( a^2 \) is \([L^2][T^{-2}]\).
- Now, dimension of \( \frac{a^2}{b} \) is:
\[ \frac{[L^2][T^{-2}]}{[L][T^{-2}]} = [L][T^{0}] = [L] \]

### Conclusion:
Since the dimension of \( c \) is \([L]\), it matches with the dimensions of \( x \), \( at \), \( bt^2 \), and \( \frac{a^2}{b} \). Thus, all options A, B, C, and D are correct.

### Final Answer:
All options (A, B, C, and D) have the same dimensions as \( c \).

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