Velocity and Acceleration — Question 6

Question 6 – Solution

Strategy. A collision requires equal position vectors at the same time, not merely intersecting geometric paths.

Step 1: Match coordinates Equality requires $$t=4-t,\qquad 2t=t+1,\qquad 3-t=2t-2.$$ The first gives $t=2$, while the second gives $t=1$. Since no single time satisfies even these two equations, the objects do not collide: $$\boxed{\text{no collision for }t\ge 0}.$$

Step 2: Velocities $$\mathbf v_1=\left\langle 1,2,-1\right\rangle,\qquad \mathbf v_2=\left\langle -1,1,2\right\rangle.$$ The velocity of object 1 relative to object 2 is $$\boxed{\mathbf v_{1/2}=\mathbf v_1-\mathbf v_2=\left\langle 2,1,-3\right\rangle}.$$

Step 3: Relative-position check $$\mathbf r_1-\mathbf r_2 =\left\langle 2t-4,t-1,5-3t\right\rangle.$$ For this vector to be zero, its first, second, and third components would require respectively $t=2$, $t=1$, and $t=5/3$. The inconsistent times independently confirm that relative position never vanishes.