A particle starts from origin at with a velocity m/s and moves in plane under action of a force which produces a constant acceleration of . If the x-coordinate of the particle at that instant is 84 m, then the speed of the particle at this time is The

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Solution

Q.
A particle starts from origin at $t = 0$ with a velocity $5 \hat{i}$ m/s and moves in $x - y$ plane under action of a force which produces a constant acceleration of $(3 \hat{i} + 2 \hat{j}) m / s^{2}$ . If the x-coordinate of the particle at that instant is 84 m, then the speed of the particle at this time is $\sqrt{α} m / s .$ The value of $α$ is ____________ . [2024]

Ans.
(673) Given, $u_{x} = 5$ $m / s, a_{x} = 3$ $m / s^{2}$ and $x = 84$ $m$

$Using, v_{x}^{2} - u_{x}^{2} = 2 a x$

$v_{x}^{2} - 25 = 2 (3) (84) \Rightarrow v_{x} = 23 m / s$

$v_{x} - u_{x} = a_{x} t \Rightarrow t = \frac{23 - 5}{3} = 6 s$

$v_{y} = 0 + a_{y} t = 0 + 2 \times (6) = 12 m / s$

$v^{2} = v_{x}^{2} + v_{y}^{2} = 23^{2} + 12^{2} = 673$

$\Rightarrow v = \sqrt{673} m / s$

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