The relation between time and distance is where and are constants. The relation between acceleration (a) and velocity (v) is

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Solution

Q.
The relation between time $' t'$ and distance $' x'$ is $t = α x^{2} + β x,$ where $α$ and $β$ are constants. The relation between acceleration (a) and velocity (v) is [2024]

1 $a = - 2 α v^{3}$

2 $a = - 2 α v^{4}$

3 $a = - 2 α v^{2}$

4 $a = - 2 α v^{5}$

Ans.
(1)

$t = α x^{2} + β x$ (differentiating wrt time)

$\frac{d t}{d x} = 2 α x + β$

$\frac{1}{v} = 2 α x + β$

(differentiating wrt time) $- \frac{1}{v^{2}} \frac{d v}{d t} = 2 α \frac{d x}{d t}$

$\frac{d v}{d t} = - 2 α v^{3}$

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