A particle is projected from the ground at an angle 600 with the horizontal with an initial speed 20ms-1. After n3sec the velocity vector of projectile will be perpendicular to the initial velocity. The value of n is____. g=10m/s2

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A particle is projected from the ground at an angle 60⁰ with the horizontal with an initial speed 20ms^-1. After $\frac{n}{\sqrt{3}} \sec$ the velocity vector of projectile will be perpendicular to the initial velocity. The value of n is____. $(g = 10 m / s^{2})$

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answer is 4.

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Initial velocity

$u = u \cos θ \hat{i} + u \sin θ \hat{j}$

Velocity after time t is

$v = u \cos θ \hat{i} + (u \sin θ - g t) \hat{j}$

Since u and $v$ are perpendicular to each other, therefore,

$\vec{u}, \vec{v} = 0$

$(u \cos θ \hat{i} + u \sin θ \hat{j}) \times [u \cos θ i + (u \sin θ - τ) \hat{j}] = 0$

$u^{2} \cos^{2} θ + u^{2} \sin^{2} θ - u \sin θ g t = 0$

$t = \frac{u}{g \sin θ} = \frac{20}{10 \times \frac{\sqrt{3}}{2}} = \frac{4}{\sqrt{3}} s$

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