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

Question 1

Skill question

For $P (t) = 214 + t 591$ and $Q (t) = 101 + t 872$ , find the time of closest approach and verify that $D (t) = P (t) - Q (t)$ at this time is perpendicular to the relative velocity.

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Question 2

Skill question

Two particles move along trajectories $r_{1} (t) = 214 + t 591$ and $r_{2} (t) = 101 + t 872$ , where $t$ is time.

Find the time $t$ at which the distance between the two particles is minimized.

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Question 3

Skill question

The question asks for the derivation of the time $t$ that minimizes the distance between two points moving at constant velocities in 3D space, and the subsequent application of this formula to a specific case.

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

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Question 4

Skill question

Two particles $P$ and $Q$ move such that their position vectors at time $t$ are given by $P (t) = 012 + t 304$ and $Q (t) = 100 + t 043$ .

Find the time $t$ at which the particles are at their point of minimum separation.

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Question 5

Skill question

Two particles $P$ and $Q$ move such that their position vectors at time $t$ are given by: $r_{P} (t) = 400 + t 122$ $r_{Q} (t) = 111 + t 213$ where $t \geq 0$ is the time in seconds.

Determine the time $t$ at which the particles are at their closest approach.

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Question 6

Skill question

The positions of two particles $P$ and $Q$ at time $t$ are given by the vectors $P (t)$ and $Q (t)$ .

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.

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Question 7

Skill question

Calculus-based optimization of distance between two moving points in 3D space.

Two points move such that their positions at time $t$ are given by $P (t) = (2 t, 2 t, 0)$ and $Q (t) = (1 - t, 2 + t, 0)$ .

Find the time $t$ when the distance between the two points is smallest.

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Question 8

Skill question

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

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Question 9

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.

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Question 10

Skill question

Determine the time $t$ at which 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.

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Question 11

Skill question

The question asks for the value of $t$ that minimizes the distance between two points moving in three-dimensional space. The paths of the points are given as vector equations.

Find the time $t$ 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.

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Question 12

Skill question

Optimization of distance between moving objects using vector equations.

Two objects follow paths given by $P (t) = 310 + t - 1 21$ and $Q (t) = - 2 41 + t 2 - 1 0$ .

Find the time $t$ at which the objects are closest to each other.

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IB Mathematics Applications & Interpretation Question Type 4: Finding the Time at Which Two Objects Are Closest Each Other Given Some Trajectory Exercises