A particle is projected with a speed u at an angle with the horiz… - Vidyadip

Question

A particle is projected with a speed u at an angle $\theta$ with the horizontal. Consider a small part of its path near the highest position and take it approximately to be a circular arc. What is the radius of this circle? This radius is called the radius of curvature of the curve at the point.

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Answer

Particle is projected with speed ‘u’ at an angle $\theta.$ At the highest pt. the vertical component of velocity is‘0’.

So, at that point, velocity $=\text{u}\sin\theta$

centripetal force $=\text{mu}^2\cos^2\Big(\frac{\theta}{\text{r}}\Big)$

At highest pt.

$\text{mg}=\frac{\text{mv}^2}{\text{r}}$

$\Rightarrow\text{r}=\frac{\text{u}^2\cos^2\theta}{\text{g}}$

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