If x=a+bt+ct2, where x is in meter and t is in second. What is the unit of b and c ?

1. Break Down the Problem

To determine the units of $b$ and $c$ in the equation $x = a + bt + ct^2$, we need to analyze the units of each term in the equation. Here, $x$ is given in meters (m) and $t$ is given in seconds (s).

2. Relevant Concepts

The equation can be treated term by term:

• The dimension of $a$ (in meters).
• The term $bt$ must also have units of meters.
• The term $ct^2$ must also have units of meters.

3. Analysis and Detail

1. Units of $a$:

   • $a$: has units of meters (m).

2. Units of $bt$:

   • For the term $bt$ to have units of meters: $[b][t] = \text{meters} \implies [b] \cdot [s] = [m]$ From this, we can solve for $[b]$: $[b] = \frac{\text{meters}}{\text{seconds}} = \text{m/s}$

3. Units of $ct^2$:

   • For the term $ct^2$ to have units of meters: $[c][t^2] = \text{meters} \implies [c] \cdot [s^2] = [m]$ From this, we can solve for $[c]$: $[c] = \frac{\text{meters}}{\text{seconds}^2} = \text{m/s}^2$

4. Verify and Summarize

• The unit of $b$ is derived as $\text{m/s}$.
• The unit of $c$ is derived as $\text{m/s}^2$.
• Therefore, both units are consistent with the equation given.

Final Answer

• The unit of $b$ is $\text{m/s}$.
• The unit of $c$ is $\text{m/s}^2$.