Geometry of motion - kinematics

Position

Classical physics describes the position of an object using three independent coordinates e.g.

()\[ \mathbf{r}_{P/O} = x\hat{i} + y\hat{j} + z\hat{k} \]

where \(\mathbf{r}_{P/O}\) is the position of point \(P\) with respect to the point of origin \(O\), \(x,~y,~z\) are magnitudes of distance along a Cartesian coordinate system and \(\hat{i},~\hat{j}\) and \(\hat{k}\) are unit vectors that describe three

import numpy as np
import matplotlib.pyplot as plt
plt.style.use('fivethirtyeight')

Velocity

The velocity of an object is the change in position, equation (), per length of time.

()\[ \mathbf{v}_{P/O} = \frac{d\mathbf{r}_{P/O}}{dt} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k} \]

Note

The notation \(\dot{x}\) and \(\ddot{x}\) is short-hand for writing out \(\frac{dx}{dt}\) and \(\frac{d^2x}{dt^2}\), respectively.

The definition of velocity in equation () depends upon the change in position of all three independent coordinates, where \(\frac{d}{dt}(x\hat{i})=\dot{x}\hat{i}\).

Note

Remember the chain rule: \(\frac{d}{dt}(x\hat{i})=\dot{x}\hat{i} + x\dot{\hat{i}}\), but \(\dot{\hat{i}}=0\) because this unit vector is not changing direction. You’ll see other unit vectors later that do change.

postion-velocity

Example - GPS vs speedometer

You can find velocity based upon postion, but you can only find changes in position with velocity. Consider tracking the motion of a car driving down a road using GPS. You determine its motion and create the position, \(\mathbf{r} = x\hat{i} +y\hat{j}\), where

\(x(t) = 4t +3\) and \(y(t) = 3t - 1\)

To get the velocity, calculate \(\mathbf{v} = \dot{\mathbf{r}}\)

\(\mathbf{v} = 4\hat{i} +3 \hat{j}\)

t = np.arange(0,5)
x = 4*t + 3
y = 3*t -1
plt.plot(x,y,'o')
plt.quiver(x,y,4,3)
plt.title('Position of car on road every 1 second'+
    '\nvelocity shown as arrow')
plt.xlabel('x-position (m)')
plt.ylabel('y-position (m)');

Speed

The speed of an object is the magnitude of the velocity,

\(|\mathbf{v}_{P/O}| = \sqrt{\mathbf{v}\cdot\mathbf{v}} = \sqrt{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}\)

Acceleration

The acceleration of an object is the change in velocity per length of time.

\(\mathbf{a}_{P/O} = \frac{d \mathbf{v}_{P/O} }{dt} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}\)

where \(\ddot{x}=\frac{d^2 x}{dt^2}\) and \(\mathbf{a}_{P/O}\) is the acceleration of point \(P\) with respect to the point of origin \(O\).

Rotation and Orientation

The definitions of position, velocity, and acceleration all describe a single point, but dynamic engineering systems are composed of rigid bodies is needed to describe the position of an object.

from IPython.core.display import SVG

SVG(filename='./images/position_angle.svg')

In the figure above, the center of the block is located at \(r_{P/O}=x\hat{i}+y\hat{j}\) in both the left and right images, but the two locations are not the same. The orientation of the block is important for determining the position of all the material points.

In general, a rigid body has a pitch, yaw, and roll that describes its rotational orientation, as seen in the animation below. We will revisit 3D motion in Module_05

from IPython.display import YouTubeVideo
vid = YouTubeVideo("li7t--8UZms?loop=1")
display(vid)

Rotation in planar motion

Our initial focus is planar rotations e.g. yaw and roll are fixed. For a body constrained to planar motion, you need 3 independent measurements to describe its position e.g. \(x\), \(y\), and \(\theta\)