We present a continuous-time collision detection algorithm for quickly detecting whether certain polynomial trajectories in time intersect with convex obstacles. The algorithm is used in conjunction with an existing multicopter trajectory generation method to achieve rapid, obstacle-aware motion planning in environments with both static convex obstacles and dynamic convex obstacles whose boundaries do not rotate. In general, this problem is difficult because the presence of convex obstacles makes the feasible space of trajectories nonconvex. The performance of the algorithm is benchmarked using Monte Carlo simulations, and experimental results are presented that demonstrate the use of the method to plan collision-free multicopter trajectories in milliseconds in environments with both static and dynamic obstacles.