In the euqation of motion, `x=at + bt^(2)`, the units of a and b are respectively

Options

A. `m/s^(2)`
B. `m/s, m/s^(2)`
C. m/s, m/s
D. none of these

Explanation

To determine the units of \( a \) and \( b \) in the equation of motion \( x = at + bt^2 \), we will analyze the equation step by step using the principle of homogeneity.

### Step 1: Identify the dimensions of each term
The equation is given as:
\[ x = at + bt^2 \]

Here, \( x \) represents the displacement, which has the dimension of length (L).

### Step 2: Analyze the term \( at \)
In the term \( at \):
- \( x \) has the dimension of length (L).
- \( t \) represents time, which has the dimension of time (T).

To maintain the homogeneity of the equation, the dimension of \( at \) must also be length (L). Therefore, we can express the dimension of \( a \) as follows:
\[ [a] \cdot [t] = L \]
\[ [a] \cdot T = L \]
Thus, rearranging gives:
\[ [a] = \frac{L}{T} \]

### Step 3: Determine the units of \( a \)
The unit of \( a \) can be expressed in terms of standard units:
- Length (L) is measured in meters (m).
- Time (T) is measured in seconds (s).

Therefore, the unit of \( a \) is:
\[ \text{Unit of } a = \frac{\text{meters}}{\text{seconds}} = \text{m/s} \]

### Step 4: Analyze the term \( bt^2 \)
Next, we analyze the term \( bt^2 \):
- The dimension of \( bt^2 \) must also equal length (L).

Thus, we can express the dimension of \( b \) as follows:
\[ [b] \cdot [t^2] = L \]
\[ [b] \cdot T^2 = L \]
Rearranging gives:
\[ [b] = \frac{L}{T^2} \]

### Step 5: Determine the units of \( b \)
Using standard units:
- Length (L) is measured in meters (m).
- Time squared (T²) is measured in seconds squared (s²).

Therefore, the unit of \( b \) is:
\[ \text{Unit of } b = \frac{\text{meters}}{\text{seconds}^2} = \text{m/s}^2 \]

### Summary of Units
- The unit of \( a \) is \( \text{m/s} \).
- The unit of \( b \) is \( \text{m/s}^2 \).

### Final Answer
The units of \( a \) and \( b \) are respectively \( \text{m/s} \) and \( \text{m/s}^2 \).

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