A question asks which direction a particle is moving in. Which vector do you use, and why not the position vector?

Use the velocity : the position vector says where the particle is , measured from the origin. The velocity says where it is heading , and the two need not point the same way: a particle can be north-east of the origin while travelling due south. The direction of the velocity may then be wanted as an angle, as a bearing, or as a vector that the velocity is parallel to.

A LevelMathsEdexcelMechanicsFlashcardsKinematicsVariable Acceleration in 2D

Variable Acceleration in 2D (Edexcel A Level Maths: Mechanics): Flashcards

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FrontUsing Calculus in 2D
How do you differentiate or integrate a vector in kinematics?

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How do you differentiate or integrate a vector in kinematics?
Work on each component separately.
To differentiate, differentiate the $i$ component and the $j$ component; to integrate, integrate each of them.
The links between displacement, velocity and acceleration are the same as in one dimension, and only the displacement changes name: in two dimensions it is usually written $r$ rather than $s$ .
True or False?
You can assume that the constant of integration in a two-dimensional kinematics problem is a single number.
False.
Integrating a vector gives a vector constant of integration, with an $i$ component and a $j$ component.
Find it by substituting a known vector and then equating the $i$ components and the $j$ components separately, which gives one equation for each part of the constant.
So, for example, integrating a velocity might give $r = t^{2} i + 2 e^{0.5 t} j + c$ , and a starting position of $(3 i - 4 j) m$ then gives $c = (3 i - 6 j) m$ .
A particle's position vector is $r$ and its displacement from where it started is $s$ . How are the two related, and why are they not the same thing?
They are related by $r = r_{0} + s$ , where $r_{0}$ is the particle's initial position vector.
They differ because $s$ is measured from wherever the particle started, while $r$ is measured from the origin, and a particle need not start at the origin.
When you integrate a velocity to find the position, $r_{0}$ is exactly what the constant of integration turns out to be.
A particle's velocity is $(3 i - 4 j) {m s}^{- 1}$ . What is its speed?
Its speed is $5 {m s}^{- 1}$ .
Speed is the magnitude of the velocity, so it comes from Pythagoras' theorem on the two components:
$|v| = \sqrt{3^{2} + {(- 4)}^{2}} = 5 {m s}^{- 1}$
The same move applied to a displacement gives the distance from the starting point; both are single numbers and neither can be negative.
A question asks which direction a particle is moving in. Which vector do you use, and why not the position vector?
Use the velocity: the position vector says where the particle is, measured from the origin.
The velocity says where it is heading, and the two need not point the same way: a particle can be north-east of the origin while travelling due south.
The direction of the velocity may then be wanted as an angle, as a bearing, or as a vector that the velocity is parallel to.

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