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Finding Average Velocity

A particle is...

Question

A particle is projected with a velocity u making an angle $θ$ with the horizontal. What is the radius of curvature of the parabola where the particle makes an angle $θ$ /2 with the horizontal?

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Solution

$_{i n} = u c o s θ^i + u s i n θ^j a n d \to v (t) = u c o s θ^i + (u s i n θ - g t)^j a t t i m e t t h e a n g l e o f t h e p r o j e c t i l e i s θ / 2 \frac{u s i n θ - g t}{u c o s θ} = t a n θ / 2 \Rightarrow u s i n θ - g t = u c o s θ t a n θ / 2 \to v (t) = u c o s θ^i + (u c o s θ t a n θ / 2)^j \Rightarrow v = \sqrt{{(u c o s θ)}^{2} + {(u c o s θ t a n θ / 2)}^{2}} \Rightarrow v = u c o s θ s e c θ / 2 R = \frac{v^{2}}{g c o s θ / 2} = \frac{u^{2} {c o s}^{2} θ {s e c}^{2} θ / 2}{g c o s θ / 2} = \frac{u^{2} {c o s}^{2} θ}{g {c o s}^{3} θ / 2}$

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