Kinematics

Description of Motion

  What is Physics?

    Overview

           Contents

Herein, you will find resources related to the following topics:

Kinematics

Starring ...

 Kinematics

    Staring ...

         Position

All the world's a stage,

and all the matter and energy are merely players,

 Kinematics

    Featuring ...

         Position and Time

\vec{r}

relative position

All positions are relative.

 Kinematics

    Starring ...

         Position & Time

\vec{r}

relative position

\Delta t
\vec{r}_\text{A}
\vec{r}_\text{B}

Motion requires a change in position

Kinematics

Displacement & Distance

 Kinematics

    Featuring ...

         Displacement & Distance

\vec{r}

relative position

\Delta t
\vec{r}_\text{A}
\vec{r}_\text{B}

The object is displaced

from A to B

along the

shown path

 Kinematics

    Featuring ...

\vec{r}

relative position

\Delta t
\vec{r}_\text{A}
\vec{r}_\text{B}

The Displacement is defined as the change in position

\Delta\vec{r} = \vec{r}_\text{B}-\vec{r}_\text{A}
\Delta\vec{r}
\Delta\vec{r}
\vec{r}_\text{B}
-\vec{r}_\text{A}
\Delta\vec{r}
\vec{r}_\text{A}

         Displacement & Distance

 Kinematics

    Featuring ...

\vec{r}

relative position

\Delta t
\vec{r}_\text{A}
\vec{r}_\text{B}

The Displacement is defined as the change in position

\Delta\vec{r} = \vec{r}_\text{B}-\vec{r}_\text{A}
\Delta\vec{r}

The magnitude of the displacement is the shortest distance between the two points, A and B.

The direction of the displacement is from the starting point, A, towards the end point, B.

         Displacement & Distance

 Kinematics

    Featuring ...

Suppose we divide the path into little infinitesimal displacements along the path

\Delta\vec{r} = \int_A^Bd\vec{r}
\vec{r}_\text{A}
\vec{r}_\text{B}
\Delta t

 Displacement

d = \int_A^B|d\vec{r}|

Distance Traveled

\Delta\vec{r}
d\vec{r}
\text{(aka $d\vec{s}$)}

         Displacement & Distance

 Kinematics

    Featuring ...

\vec{r}_\text{A}
\vec{r}_\text{B}
\Delta t

         Displacement & Distance

In summary, distance is a path dependent scalar quantity, whereas displacement is a path independent vector quantity.

{\small\int} |d\vec{r}|
\Delta\vec{r}

Kinematics

Velocity & Speed

 Kinematics

    Featuring ...

\vec{r}_\text{A}
\vec{r}_\text{B}
\Delta t

         Velocity & Speed

The average velocity during some segment of motion is defined as

{\small\int} |d\vec{r}|
\Delta\vec{r}
\vec{v}_\text{avg} = \frac{\Delta \vec{r}}{\Delta t}

The "state of motion" is represented by a vector physical quantity denoted the Velocity

 Kinematics

    Featuring ...

         Velocity

Consider the component of the motion in some direction:

+x
-x
\vec{v}_\text{avg-x} = \frac{\Delta \vec{x}}{\Delta t}

The Average Velocity is defined as:

 Kinematics

    Featuring ...

         Instantaneous Velocity

Consider the component of the motion in some direction:

+x
-x
\vec{v}(t) = \lim_{\Delta t\to 0} \frac{\Delta \vec{x}}{\Delta t}=\frac{d\vec{x}}{dt}
\vec{v}_x(t) = \frac{d \vec{x}}{d t}

The Instantaneous Velocity is defined:

 Kinematics

    Featuring ...

         Instantaneous Velocity

Consider the component of the motion in some direction:

+x
-x
\vec{v}_x(t) = \frac{d \vec{x}}{d t}

The Instantaneous Velocity is defined:

 Kinematics

    Featuring ...

         Instantaneous Velocity

\vec{v}_x(t) = \frac{d \vec{x}}{d t}
\vec{v}_x(t)\ dt = {d \vec{x}}
\Delta x = \int_{t_0}^{t} \vec{v}_x(t)\ dt

Kinematics

Acceleration

 Kinematics

    Featuring ...

         Acceleration

\vec{a}_\text{avg-x} = \frac{\Delta \vec{v}_x}{\Delta t}
\vec{a}_x(t) = \frac{d \vec{v}_x}{d t}

Kinematics

Motion w/ Constant Acceleration

 Kinematics

    Motion in 1-D

         Motion w/ Constant Acceleration

\vec{a}(t) = \frac{d \vec{v}}{d t}

 Kinematics

    Motion in 1-D

         Motion w/ Constant Acceleration

\vec{v}(t) = \frac{d \vec{x}}{d t}

 Kinematics

    Motion in 1-D

         Motion w/ Constant Acceleration

Kinematic Equations

Kinematic variables

 Kinematics

    Motion in 1-D

         Motion w/ Constant Acceleration

Kinematic Equations

Kinematic variables

Kinematics

Freefall 

 Kinematics

    Motion in 1-D

         Freefall

 If a bowling ball and a feather are dropped from the same height, would they reach the ground at the same time?

Independent of

the direction of motion!!!

\vec{a}_\text{\tiny due to gravity alone} = g \text{ \tiny downwards}

 Kinematics

    Motion in 1-D

         Freefall

Independent of

the direction of motion!!!

\vec{a}_\text{\tiny due to gravity alone} = g \text{ \tiny downwards}

"During the final minutes of the third extravehicular activity [of the Apollo 15 mission,] a short demonstration experiment was conducted. A heavy object (a 1.32-kg aluminum geological hammer) and a light object (a 0.03-kg falcon feather) were released simultaneously from approximately the same height ... and were allowed to fall to the surface. Within the accuracy of the simultaneous release, the objects were observed to undergo the same acceleration and strike the lunar surface simultaneously, which was a result predicted by well-established theory, but a result nonetheless reassuring considering both the number of viewers that witnessed the experiment and the fact that the homeward journey was based critically on the validity of the particular theory being tested."

Watching the footage, I estimate the time interval from 'release' to 'touch ground' to be around ___ seconds or so, and I estimate the displacement to be about ____ or so, downwards.

Using these values, I can estimate the moon's gravitational acceleration to be ____ m/s^2, which is _____% away from the accepted value of _____ m/s^2.

Kinematics

in 2+ Dimensions 

 Kinematics

    Motion in 2+ D

         In general

Describe the motion using vector mathematics.

\vec{r}_\text{A}
\vec{r}_\text{B}
\Delta\vec{r}
Physical Quantity      
Displacement
Velocity
Acceleration
Time Interval
\hat{i}
\hat{j}
\hat{k}
\Delta t
\Delta\vec{r}
\Delta\vec{x}
\Delta\vec{y}
\Delta\vec{z}
\vec{a}
\vec{a}_x
\vec{a}_y
\vec{a}_z
\vec{v}
\vec{v}_x
\vec{v}_y
\vec{v}_z
\vec{v}_A
\vec{v}_B
\vec{v}

 Kinematics

    Motion in 2+ D

         In general

Independence of the components

\Delta \vec{x}=\vec{v}_{0x} t+\tfrac{1}{2}\vec{a_x}t^2
\Delta \vec{y}=\vec{v}_{0y} t+\tfrac{1}{2}\vec{a_y}t^2
\Delta \vec{r}=\vec{v}_0 t+\tfrac{1}{2}\vec{a}t^2
\left(\Delta \vec{r}=\vec{v}_0 t+\tfrac{1}{2}\vec{a}t^2\ \right)_x
\left(\Delta \vec{r}=\vec{v}_0 t+\tfrac{1}{2}\vec{a}t^2\ \right)_y

Describe the motion using vector mathematics.

 Kinematics

    Motion in 2-D

         Projectile Motion

An object under the influence of gravity alone, experiences the acceleration:

Independent of

the direction of motion!!!

\vec{a}_\text{\tiny due to gravity alone} = g \text{ \tiny downwards}

 Kinematics

    Motion in 2+ D

        Uniform Circular Motion

\hat{i}
\hat{j}
\hat{k}
\vec{v}
\vec{v}_x
\vec{v}_y
\vec{v}_z
\vec{v}_1
\vec{v_2}
\Delta\vec{r}

 Kinematics

    Motion in 2-D

         Uniform Circular Motion

Imagine the motion of an object on a circular arc of radius R with constant speed v

towards the center of the motion

\vec{a}_\text{centripetal} = \frac{v^2}{R}