Hidden Content Control Panel: Reveal All | Hide All

S1.2 Kinematics in two (or three) dimensions

[A] Position and displacement vectors

[B] The velocity vector

Key Point 1.5

The velocity is the time rate of change of the position vector; it is a vector:
\[ \vec{v} = \frac{d \vec{r}}{dt} \hspace*{1cm}\mbox{{\rm In component form:}}\hspace*{0.3cm} v_x = \frac{dx}{dt} \hspace*{0.5cm} v_y = \frac{dy}{dt} \]

Commentary

Reveal
Hide

Commentary

Write:

\[ \vec{r}= x\hat{i} + y\hat{j} \]
Differentiate:

\[ \frac{d \vec{r}}{dt} = \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{\frac{d x}{dt}\hat{i} + \frac{dy}{dt} \hat{j}} \]

Help?

Help?

Combining ‘vectors’ and ‘differentiation’ may feel like a ‘bridge too far’. But just take it step by step.
  • Realise that $\vec{r}$ is simply short for

    \[ x\hat{i} +y \hat{j} \]

  • Recognise that the things that depend on time are the scalar quantities x and y; the unit vectors î and $\hat{j}$ are ‘constant’ (independent of time)...they are the signposts that define our coordinate axes.
  • Using the fact that î is constant we can write
    \[ \frac{d}{dt} (x\hat{i}) = \hat{i} \frac{dx}{dt} \]
    since the unit vector can be pulled out from the differentation (which is only interested in things that do depend on time.)
  • Doing the same with the $\vec{j}$ term gives the result.
 

Equate to:

\[ \vec{v}= \xmlInlineElement[\xmlAttr{}{target}{slides}]{http://www.ph.ed.ac.uk/aardvark/NS/aardvark-latex}{aardvark:reveal}{v_x\hat{i} + v_y\hat{j}} \]
Then
\[ v_x\hat{i} + v_y\hat{j} = \frac{d x}{dt}\hat{i} + \frac{dy}{dt} \hat{j} \]
Take the dot product of each side with î:

$$v_x= \frac{d x}{dt}$$

Help?

Help?

You need to remember the rules about the dot products of unit vectors (S0.4):
\[ \hat{i}\cdot \hat{j}=0 \hspace{1cm}\mbox{\rm and}\hspace{1cm}\hat{i}\cdot\hat{i}=1 \]
 

Dotting with $\hat{j}$ gives the vy equation.

 

Key Point 1.6

The velocity vector is always tangential to the particle path.
AInstantaneous and average velocities

[C] The acceleration vector

Key Point 1.7

The acceleration is the time rate of change of the velocity; it is a vector:
\[ \vec{a} = \frac{d \vec{v}}{dt} \hspace*{1cm}\mbox{{\rm In component form:}}\hspace*{0.3cm} a_x = \frac{dv_x}{dt} \hspace*{0.5cm} a_y = \frac{dv_y}{dt} \]

[D] Constant acceleration equations

TAcceleration of a Ball Travelling in an arc
TMonkey and Hunter

Learning Resources

Textbook: HRW Chapter 4.1-4
Self-Test Questions: