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

Options

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

Answer

To determine 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 can follow these steps:

### Step 1: Understand the equation
The equation \( x = at + bt^2 \) represents the distance traveled by a body. Here, \( x \) is in kilometers (km) and \( t \) is in seconds (s).

### Step 2: Analyze the terms
Since both terms on the right-hand side of the equation must have the same units as \( x \) (which is kilometers), we need to analyze the units of each term.

### Step 3: Determine the units of \( a \)
The first term is \( at \). Since \( t \) is in seconds, we can express the units of \( a \) as follows:
\[ \text{Units of } at = \text{Units of } a \times \text{Units of } t = \text{Units of } a \times \text{s} \]
To ensure that this term has units of kilometers, we set:
\[ \text{Units of } a \times \text{s} = \text{km} \]
Thus, the units of \( a \) can be determined as:
\[ \text{Units of } a = \frac{\text{km}}{\text{s}} \quad (\text{kilometers per second}) \]

### Step 4: Determine the units of \( b \)
Now, we look at the second term, \( bt^2 \). The units can be expressed as:
\[ \text{Units of } bt^2 = \text{Units of } b \times \text{Units of } t^2 = \text{Units of } b \times \text{s}^2 \]
Again, to ensure that this term also has units of kilometers, we set:
\[ \text{Units of } b \times \text{s}^2 = \text{km} \]
From this, we can solve for the units of \( b \):
\[ \text{Units of } b = \frac{\text{km}}{\text{s}^2} \quad (\text{kilometers per second squared}) \]

### Conclusion
Thus, the units of \( b \) are \( \text{km/s}^2 \).

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