Engineer Physics/Motion/Linear motion

Linear motion refers to any motion follows a straight inclined line at an inclined angle without changing it's direction

Characteristics

Acceleration of the motion can be calculated as shown below

$a={\frac {\Delta v}{\Delta t}}={\frac {v-v_{o}}{t-t_{o}}}$

$\Delta v=a\Delta t$

Speed of the motion

$v=v_{o}+a\Delta t$

$v_{o}=v-a\Delta t$

$\Delta t={\frac {v-v_{o}}{a}}$

Distance of the motion

$s=v_{o}\Delta t+{\frac {\Delta v}{2}}\Delta t=[v_{o}+{\frac {\Delta v}{2}}]\Delta t=[v_{o}+{\frac {a\Delta t}{2}}]\Delta t=[v-{\frac {a\Delta t}{2}}]\Delta t={\frac {v^{2}-v_{o}^{2}}{2a}}$

Speed's square

$v^{2}=v_{o}^{2}+2as$

a = 0 , when acceleration is zero

$a={\frac {\Delta v}{\Delta t}}={\frac {v-v_{o}}{t-t_{o}}}=0$

$v=v_{o}$

$s=v_{o}t$

a = constant , When acceleration is a constant

$a=-g$

$v=-gt$

$s=(-gt)\times t=-gt^{2}$

a , When acceleration is not zero

$a={\frac {\Delta v}{\Delta t}}={\frac {v-v_{o}}{t-t_{o}}}$

$v=v_{o}+a\Delta t$

$s=v_{o}\Delta t+{\frac {\Delta v}{2}}\Delta t=[v_{o}+{\frac {\Delta v}{2}}]\Delta t=[v_{o}+{\frac {a\Delta t}{2}}]\Delta t=[v-{\frac {a\Delta t}{2}}]\Delta t={\frac {v^{2}-v_{o}^{2}}{2a}}$

$a={\frac {\Delta v}{\Delta t}}={\frac {v-0}{t-0}}={\frac {v}{t}}$

$v=at$

$s={\frac {1}{2}}vt$

Mathematical formula

Inclined linear motion

Characteristics	Symbol	Formula	Unit
Acceleration	$a$	${\frac {\Delta v}{\Delta t}}={\frac {v-v_{o}}{t-t_{o}}}$	m/s2
Speed	$v$	$v_{o}+a\Delta t$	m/s
Distance	$s$	$\Delta t(v_{o}+\Delta v)=\Delta t(v_{o}+a\Delta t)=\Delta t(v-a\Delta t)={\frac {v^{2}-v_{o}^{2}}{2a}}$	m
Force	$F$	$ma=m{\frac {\Delta v}{\Delta t}}$	N
Work	$W$	$Fs=F\Delta t(v_{o}+\Delta v)$	N m
Energy	$E$	${\frac {W}{t}}=F(v_{o}+\Delta v)$	N m/s

Horizontal linear motion

Tính Chất Chuyển Động	Ký Hiệu	Công Thức	Đơn vị
Gia tốc	$a$	${\frac {v_{o}}{t}}$	m/s2
Vận tốc	$v$	$v_{o}$	m/s
Đường dài	$s$	$v_{o}t$	m
Lực	$F$	$m{\frac {v_{o}}{t}}$	N
Năng lực	$W$	$Fv_{o}t$	N m
Năng lượng	$E$	$Fv_{o}$	N m/s

Vertical linear motion

Tính Chất Chuyển Động	Ký Hiệu	Công Thức	Đơn vị
Gia tốc	$a$	$-g$	m/s2
Vận tốc	$v$	$-gt$	m/s
Đường dài	$s$	$-gt^{2}$	m
Lực	$F$	$mg$	N
Năng lực	$W$	$mgh$	N m
Năng lượng	$E$	${\frac {mgh}{t}}$	N m/s