Physics Course/Motion/Linear Motion

Linear Motion refers to any motion moving in a straight line without changing it's direction

Characteristics of linear motion

Any Linear Motion travels with different speed at different time v₁ at t₁ and v at t

The Change in Speed

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

The Change in Time

$\Delta t=t-t_{o}$

The ratio of Change in Speed over Change in Time gives the Accelerarion of the motion

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

$\Delta v=a\Delta t$

From above, speed of motion

$v=v_{o}+a(t-t_{1})=v_{o}+a\Delta t$

$v_{o}=v-a(t-t_{1})=v-a\Delta t$

The distance of motion can be calculated as the area under graph

$s=\Delta t(v_{o}+{\frac {\Delta v}{2}})=\Delta t(v-{\frac {\Delta v}{2}})$

$s=\Delta t(v_{o}+{\frac {a\Delta t}{2}})=\Delta t(v-{\frac {a\Delta t}{2}})$

$s=({\frac {v-v_{o}}{a}})({\frac {2v_{o}+v-v_{o}}{2}})={\frac {v^{2}-v_{o}^{2}}{2a}}$

From above

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

Types of Linear motion

Horizontal linear motion - Linear Motion with zero acceleration a=0

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

$v=v_{o}$

$s=v_{o}t$

Linear Motion with zero acceleration a=0 represents a horizontal linear motion, straight line motion moves in the horizonral direction

Vertical linear motion - Linear Motion with constant acceleration a=g

$a=-g$

$v=-gt$

$s=-gt^{2}$

Linear Motion with constant acceleration a=-g represents a vertical linear motion, straight line motion moves in the vertical downward direction

Inclined linear motion - Linear Motion with non zero acceleration a≠0

Case 1

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

$v=v_{o}+{\frac {\Delta v}{2}}=v_{o}+{\frac {a\Delta t}{2}}$

$s=\Delta t(v_{o}+{\frac {\Delta v}{2}})=\Delta t(v-{\frac {\Delta v}{2}})$

$s=\Delta t(v_{o}+{\frac {a\Delta t}{2}})=\Delta t(v-{\frac {a\Delta t}{2}})$

$s={\frac {v^{2}-v_{o}^{2}}{2a}}$

Case 2

$t_{o}=0$

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

$v=v_{o}+at$

$s=\Delta t(v_{o}+{\frac {\Delta v}{2}})=\Delta t(v-{\frac {\Delta v}{2}})$

$s=\Delta t(v_{o}+{\frac {a\Delta t}{2}})=\Delta t(v-{\frac {a\Delta t}{2}})$

$s={\frac {v^{2}-v_{o}^{2}}{2a}}$

Case 3

$v_{o},t_{o}=0$

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

$v=at$

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