Question Type 4: Finding the time at which two objects are closest each other given some trajectory Exercises

Question 1

Skill question

Two particles move along $P(t)=(0,0,0)+t(1,2,2)$ and $Q(t)=(1,1,1)+t(2,1,3)$ . Determine the time of closest approach.

Question 2

Skill question

Two points move with $P(t)=(2t,2t,0)$ and $Q(t)=(1-t,2+t,0)$ . Find the time when the distance between them is smallest.

Question 3

Skill question

Find the time when the distance between $P(t)=(1,2,3)+t(4,0,1)$ and $Q(t)=(2,0,5)+t(1,3,2)$ is minimized.

Question 4

Skill question

Two particles move along trajectories $P(t)=(2,1,4)+t(5,9,1)$ and $Q(t)=(1,0,1)+t(8,7,2)$ . Find the time $t$ at which the distance between them is minimized.

Question 5

Skill question

Two objects follow $P(t)=(3,1,0)+t(-1,2,1)$ and $Q(t)=(-2,4,1)+t(2,-1,0)$ . At what time are they closest?

Question 6

Skill question

Find the time of minimum separation for $P(t)=(0,1,2)+t(3,0,4)$ and $Q(t)=(1,0,0)+t(0,4,3)$ .

Question 7

Skill question

For the trajectories $P(t)=(1,0,0)+t(3,4,0)$ and $Q(t)=(0,3,0)+t(4,-3,0)$ , find the time of minimum separation and the minimum distance.

Question 8

Skill question

Determine the time and the corresponding distance when $P(t)=(0,0,1)+t(1,1,1)$ and $Q(t)=(1,2,3)+t(-1,2,0)$ are closest.

Question 9

Skill question

At what time do the moving points $P(t)=(2,2,2)+t(1,-1,2)$ and $Q(t)=(3,0,1)+t(-2,3,1)$ attain their minimum separation?

Question 10

Skill question

Find the time at which $P(t)=(1,1,0)+t(2,3,4)$ and $Q(t)=(0,2,3)+t(5,1,-1)$ are closest.

Question 11

Skill question

For $P(t)=(2,1,4)+t(5,9,1)$ and $Q(t)=(1,0,1)+t(8,7,2)$ , find the time of closest approach and verify that $D(t)=P(t)-Q(t)$ at this time is perpendicular to the relative velocity.

Question 12

Skill question

Derive the formula for the time $t$ minimizing the distance between $P(t)=P_0+tv$ and $Q(t)=Q_0+tw$ , then apply it to $P_0=(2,1,4),v=(5,9,1)$ and $Q_0=(1,0,1),w=(8,7,2)$ .