Kinematics

This page is all about kinematics for vectors. You can describe the position of a moving object using vectors. This is very much like the equation of a straight line except that the parameter is t for time. The equation below is an example: the objects initial position is (2,3) and the velocity is given as the vector \(\left( \begin{matrix} -1 \\ 4 \end{matrix} \right) \).

\(\textbf{r}=\left( \begin{matrix} 2 \\ 3 \end{matrix} \right) +t\left( \begin{matrix} -1 \\ 4 \end{matrix} \right) \)


Key Concepts

On this page, you should learn about

  • applications of vector lines to kinematics
  • speed and velocity
  • collisions

Essentials

The following videos will help you understand all the concepts from this page

Torpedo Challenge

This game will help you get a good understand kinematics with vectors. The parameter t is controlled by the slider

  • The aim of the game is to fire a torpedo to sink the submarine.
  • The submarine is initially positioned at (1,1) and you are at (3, 0)
  • Once the submarine is launched, it takes 1 second to calculate and enter the direction of your torpedo shot.
  • What direction should you fire your torpedo? Press play to see what happens.
  • Change the direction of the torpedo by entering different numbers in red.
  • Can you make them collide after 1 second, 2 seconds, 3 seconds?
  • How do you make the bomb travel faster?
  • You can recreate a different situation if you click ‘change the submarine’.

Velocity Vectors

In the following video we are going to look we will try and gain a conceptual understanding of velocity vectors. One of the key ideas of this topic is to decide if objects collide. It is not enough to consider if their paths cross. We need to think about whether they occupy the same position at the same moment in time.


To get you started, you might like to play this game to give you an idea about what is going on. Try to hit the submarine with the torpedo!


Now let's consider the example below:

A submarine is initially positioned at (0, 5) travels with velocity \(\left( \begin{matrix} 4 \\ -3 \end{matrix} \right) \\ \)ms-1 .

One second later a torpedo is fired from (3, 0) with velocity \(\left( \begin{matrix} 5 \\ 1 \end{matrix} \right) \\ \)ms-1 .

Does the torpedo manage to shoot the submarine?

Notes from the video

 

 

Summary

Print from here

Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

Exam-style Questions

Question 1

During an air show, two planes, A and B, perform a manoeuvre in which their paths cross in a near miss. The two planes are flying at the same altitude.

\(\textbf{ r }_{ A }=\left( \begin{matrix} 150 \\ 320 \end{matrix} \right) +t\left( \begin{matrix} 200 \\ 300 \end{matrix} \right) \)

\(\textbf{ r }_{ B }=\left( \begin{matrix} 875 \\ 110 \end{matrix} \right) +t\left( \begin{matrix} -100 \\ 400 \end{matrix} \right) \)

t = time in seconds. Distances are given in metres.

a) Show that the two planes cross paths, but the planes do not collide

b) Find the distance between the planes when t = 0.

c) Show that the distance d between A and B at any time t can be given by the expression

d = \(\sqrt { 100000{ t }^{ 2 }-477000t+569725 } \)

d) To the nearest metre, find the closest distance that the two planes get to one another.

 

Hint

Full Solution

 

Question 2

Distances in this question are given in metres

Two brothers, Orville and Wilbur are testing their model airplanes. The position of Orville's airplane t seconds after taking off from ground level is given by

\(\textbf{ r }=\left( \begin{matrix} 12 \\ -19\\0 \end{matrix} \right) +t\left( \begin{matrix} -4 \\ 4\\3 \end{matrix} \right) \)

a) Find the height of the plane after 4 seconds.

b) Wilbur's airplane takes off after Orville's airplane s seconds after taking off is given by

\(\textbf{ r }=\left( \begin{matrix} -26 \\ 25\\0 \end{matrix} \right) +s\left( \begin{matrix} 2 \\ -4\\8 \end{matrix} \right) \)

Find the angle between the two paths.

c) The two airplanes collide at (-20,13,24). How long after Orville’s airplane takes off does Wilbur’s airplane take off?

d) Find the speed of the two airplanes at the moment of the collision.

 

Hint

Full Solution

 

Question 3

All distances in this question are in km.

An interceptor missile, M1 is positioned at the origin. A missile, M2 is launched from (-20,7) with velocity \(\left( \begin{matrix} 3 \\ 1 \end{matrix} \right) \) kms-1 . M1 is capable of twice the speed of M2 . How many seconds later must the interceptor missile, M1 be launched if it is to travel the shortest possible distance?

Hint

Full Solution

 

MY PROGRESS

How much of Kinematics have you understood?