Question 6.Q.3: Complete circles How large must the initial speed be for P t...

For complete circles to be executed, it is necessary (and sufficient) that v > 0 and N ≥ 0 at all times, that is,

u^{2}>2 g b(1-\cos \theta) \quad \text { and } \quad u^{2} \geq g b(2-3 \cos \theta)

for all values of θ. For these inequalities to hold for all θ, the speed u must satisfy u² > 4gb and u² ≥ 5gb respectively. Since the second of these conditions implies the first, it follows that P will execute complete circles if u² ≥ 5gb.