1.A particle is projected from the ground with speed $u$ at angle $\theta$. Find the radius of curvature of its path at the highest point and at the point of projection.
Sol: At top: $v = u\cos\theta, a_n = g \implies R = \frac{(u\cos\theta)^2}{g} = \frac{u^2 \cos^2\theta}{g}$. At projection: $v = u, a_n = g\cos\theta \implies R = \frac{u^2}{g\cos\theta}$.
2.A boy aims a gun directly at a target hanging at height $H$ at distance $D$. The target is released at the exact moment the gun is fired. Prove that the bullet hits the target regardless of initial bullet velocity $u$.
Sol: Time for bullet to reach horizontal distance $D$: $t = \frac{D}{u\cos\theta}$. Vertical displacement of bullet $y_b = (u\sin\theta)t - \frac{1}{2}gt^2 = D\tan\theta - \frac{1}{2}gt^2 = H - \frac{1}{2}gt^2$. Vertical position of falling target $y_t = H - \frac{1}{2}gt^2$. Since $y_b = y_t$, collision occurs!
