(8) Linear Kinematics

Kinematic quantities

• Linear Position

• Linear Distance and Displacement

• Linear speed and velocity

  • average vs instantaneous

• Linear acceleration

Vector and Scalar

• Tip to tail method (adding multiple vectors)

• Vector Resolution and Composition (how to utilize vectors)

• Component method

Linear Position

• Linear location of object of interest at a given instant

• Vector: Magnitude + Direction

• Coordinates: r= [x, y] (r is the position vector)

• Unit: m (meter)

Linear Distance

• length of the path

  • how long/far

• Scalar: Magnitude only

• Unit: M

Linear Displacement

• change in linear position

  • net effect of motion

• Vector

• Unit: m

d= delta r= r₂ — r₁ = [x₂ - x₁, y₂ - y₁]

Tip to tail method- Purpose

• used in vector ADDITION

• to find the resultant(sum/net)

tip to tail- procedure

• connect all vectors tip to tail

• resultant(sum or net): the vector from the tail of the first vector to the tip of the last vector

• connection sequence is not important

Basic Trigonometric functions

• Right triangle- hypotenuse, opposite leg, adjacent leg, theta is the angle of interest

• Basic trig functions

  • sine (theta)= opposite/hypotenuse

  • cosine(theta)= adjacent/hypotenuse

  • tangent(theta)= opposite/adjacent or sin/cos

Vector resolution (decomposition)

Breaking down a vector into components

• v=v_x+v_y

• use hypotenuse and angle to find length of opposite and adjacent legs

• adj=hyp*cos(angle)

• opp=hyp*sin(angle)

Vector Composition

composing a vector from components

• v=v_x+v_y

• use adj and opp to find hyp and angle

  • use Pythagorean theorem to find hyp

  • use arctan(opp/adj) to find theta

Directions relative to the cardinal directions

Reminder- the direction after Due is the side it starts on

Example of directions

The component method

vectors can be described in components

• v=[v_x, v_y] = [vcosΘ, vsinΘ]

  • Θ is the direction angle (measured from the X-axis)

• Components

  • v_x,v_y = + or - number

  • sign= direction

  • numeric value: magnitude

v= v_x+v_y

Component method example

Vector addition

A= [A_x, A_y]

B=[B_x, B_y]

A+B= [A_x, A_y] + [B_x, B_y] = [A_x + B_x, A_y + B_y]

Example of vector addition

Example 2 of vector addition

Linear speed

• how fast an object moves

• linear distance/elapsed time v(bar)= d/𐤃t or v=d/dt

  • v (bar)= average speed

  • v= instantaneous speed

  • dt= infinitesimal duration

• scalar quantities

• unit: m/s

Linear velocity

• rate of change in linear position

  • rate of linear displacement

  • how fast an object moves in which direction

• linear displacement/elapsed time

  • v(bar)=𐤃r/𐤃t= d/𐤃t v=dr/dt=d/dt

  • dr: displacement during dt

• vector

• unit: m/s

CONT. Linear velocity

• direction of velocity=direction of displacement

• components

  • v=[v_x , v_y] = [ d_x/dt , d_y/dt]

• direction of the component velocity

  • positive: rightward/upward motion

  • negative: Leftward/downward motion

Average vs. Instantaneous

Length of time interval

• dt (infinitesimal 𐤃t) —→ instant

• v(bar)= d//𐤃t v₁= d₁/dt₁

Instantaneous v reflects the actual motion

• instant. speed = magnitude of instant velocity

constant velocity motion

• v (bar)=v

Zero velocity

• Average v=0

  • no displacement=no net motion

• instantaneous v=0

  • zero instant speed= not moving

  • constant position

basically average v can move around but it then goes back to the original point and shows there was no displacement. instantaneous does not have enough time to move so you stay in the same spot

EX a swimmer

EX another swimmer

Velocity in cyclic movement

v= Cycle Length * Cycle Frequency(rate)

• running: Stride Length (m/stride) * Stride Freq (stride/s)

• Swimming: Stroke length (m/stroke) * Stroke F (stroke/s)

Linear Acceleration

Acceleration

• rate of change in linear speed or velocity

Scalar acceleration

• change in speed/ elapsed time

  • a(bar)= 𐤃v/𐤃t= v₂-v₁/t₂-t1

  • a= dv/dt = v₂-v₁/t₂-t₁

• positive acceleration= speeding up

• negative acceleration= slowing down

Vector acceleration

• change in velocity/elapsed time

  • a(bar)= 𐤃v/ 𐤃t= v₂-v₁/ 𐤃t a= dv/dt=v₂-v₁/ dt

• Components

  • a=[a_x , a_y]= [dv_x/dt , dv_y/dt]

Vector acceleration continued

• Positive component acceleration

  • speed up of positive velocity

  • slow down of negative velocity

  • a_x>0 a_y>0

• Negative component acceleration

  • slow down of positive velocity

  • speed up of negative velocity

  • a_x<0 a_y<0

projectile motion

a=[a_x , a_y]

units in m/s²

horizontal velocity stays the same while v_y slows down and speeds up

zero acceleration

• zero average a= no net change in velocity (initial v=final v)

• zero instantaneous a= constant velocity

a(bar)= 𐤃v/𐤃t=v₂-v₁/ 𐤃t= 0 a= dv/dt=v₂-v₁/ dt= 0

𐤃v=0 and dv=0

causes of acceleration

• changes in magnitude of velocity( speed)

• change in direction of velocity

(tangential acceleration is the straight lines and radial acceleration is the other one)