If ` x= at + bt^(2)` , where `x` is the distance travelled by the body in kilometer while `t` is the time in seconds , then find the units of `b`.

Options

A. `km//s`
B. `km-s`
C. `km//s^(2)`
D. `km-s^(2)`

Explanation

To find the units of \( b \) in the equation \( x = at + bt^2 \), where \( x \) is the distance in kilometers and \( t \) is the time in seconds, we will follow these steps:

### Step 1: Identify the units of the variables
- Distance \( x \) is given in kilometers (km).
- Time \( t \) is given in seconds (s).

### Step 2: Analyze the equation
The equation \( x = at + bt^2 \) consists of three terms: \( x \), \( at \), and \( bt^2 \). For the equation to be valid, all terms must have the same units.

### Step 3: Determine the units of \( at \)
The term \( at \) has the unit of \( a \) multiplied by the unit of \( t \):
- The unit of \( t \) is seconds (s).
- Let the unit of \( a \) be \( [a] \).

Thus, the unit of \( at \) is:
\[ [a] \cdot \text{s} \]

### Step 4: Set the units of \( at \) equal to the units of \( x \)
Since \( x \) has units of kilometers (km), we can write:
\[ [a] \cdot \text{s} = \text{km} \]

From this, we can express the unit of \( a \):
\[ [a] = \frac{\text{km}}{\text{s}} \]

### Step 5: Determine the units of \( bt^2 \)
The term \( bt^2 \) has the unit of \( b \) multiplied by the unit of \( t^2 \):
- The unit of \( t^2 \) is seconds squared (s²).
- Let the unit of \( b \) be \( [b] \).

Thus, the unit of \( bt^2 \) is:
\[ [b] \cdot \text{s}^2 \]

### Step 6: Set the units of \( bt^2 \) equal to the units of \( x \)
Since \( x \) also has units of kilometers (km), we can write:
\[ [b] \cdot \text{s}^2 = \text{km} \]

### Step 7: Solve for the units of \( b \)
To find the unit of \( b \), we rearrange the equation:
\[ [b] = \frac{\text{km}}{\text{s}^2} \]

### Conclusion
The units of \( b \) are kilometers per second squared (km/s²).

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