If $x = at + bt^2$, where $x$ is the distance travelled by the body in kilometres while $t$ is the time in sec

Question

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

Accepted Answer

1. **Principle of Homogeneity:** According to the principle of homogeneity of dimensions, physical quantities can only be added or subtracted if they have the same dimensions. This means that in the equation $x = at + bt^2$, each term on the right-hand side ($at$ and $bt^2$) must have the same physical dimensions as the term on the left-hand side ($x$) .
2. **Dimensional Analysis:** 
   - The quantity $x$ represents distance, so its unit is kilometres (km) and its dimension is Length [L].
   - The quantity $t$ represents time, so its unit is seconds (s) and its dimension is Time [T].
   - For the term $bt^2$ to have the same dimensions as $x$, the units must satisfy the equation:
     $$ 	ext{Unit}(b) 	imes 	ext{Unit}(t)^2 = 	ext{Unit}(x) $$
     $$ 	ext{Unit}(b) 	imes 	ext{s}^2 = 	ext{km} $$
3. **Calculation:** Solving for the unit of $b$:
     $$ 	ext{Unit}(b) = \frac{	ext{km}}{	ext{s}^2} $$
4. **Conclusion:** The unit of $b$ is km/s$^2$.

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