If two objects travel through space along two different curves, it\'s often impo
ID: 2875370 • Letter: I
Question
If two objects travel through space along two different curves, it's often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of two particles are given by the vector functions r_1(t) = (t^2, 15t - 54, t^2) r_2(t) = (7t - 6, t^2, 10t - 24) for t Greaterthanorequalto 0. Find the values of t at which the particles collide. (Enter your answers as a comma-separated list. If an answer does not exist, enter ONE.) t = Draw the projections of the curve on the three coordinate planes. Use these projections to help sketch the curve. r(t) = (t, sin t, 2 cos t)Explanation / Answer
If the two particle collide, then there will be a value of t such that the i, j, and k components for both r1 and r2 are equal.
comparing the i components
=> t^2=7t-6
=> t = 1 and t=6
comparing the j components
=> 15t-54 = t^2
=> t= 6 and t = 9
comparing the k components
=> t^2=10t-24
=> t= 4 and t=6
since t= 6 is common in the above three equations
=> the paticles collide at t= 6 seconds