Linear Motion

Definition

Linear motion is motion along a straight line. It can be described mathematically using only one spatial dimension. The motion of an object along a line can be described by its position x, which varies with time t.

There are two types of linear motion:
- uniform linear motion with constant velocity or zero acceleration
- non uniform linear motion with variable velocity or non-zero acceleration

---

Displacement

Displacement is the shortest distance from the initial position x1 to the final position x2 of an object. Linear motion is a motion in a single dimension, the distance traveled by an object in particular direction is the same as displacement. The SI unit of displacement is the meter [m].
The displacement from the initial position to the final position of the object can be described as:

Δx=x2−x1[m]
where:
x1 - initial position of the object
x2 - final position of the object

---

Average velocity

Velocity is defined as the rate of change of displacement with respect to time. The SI unit of velocity is the meter per second ms[−1]. The average velocity is the ratio of total displacement Δx taken over time interval Δt.
The average velocity from the initial position to the final position of the object over time can be described as:
vav=

Δx

Δt

=

x2−x1

t2−t1

[

m

s

]
where:
x1 - initial position of the object
x2 - final position of the object
t1 - the time at which the object was at position x1
t2 - the time at which the object was at position x2

---

Instantaneous velocity

The instantaneous velocity is the measure of velocity of an object at a particular moment. Instantaneous velocity can be found by differentiating the displacement with respect to time.
The instantaneous velocity of an object with displacement in time can be described as:
vin=

lim

Δt→0

Δx

Δt

=

dx

dt

[

m

s

]
where:
Δx - displacement
Δt - time

---

Acceleration

Acceleration is the rate at which the velocity of an object changes with time. The SI unit of acceleration is the meter per square second m/s2. Acceleration is the ratio of total velocity change Δvtaken over time interval Δt.

a=

lim

Δt→0

Δv

Δt

=

dv

dt

[

m

s2

]
where:
Δv - velocity change
Δt - time change

The average acceleration is defined as the rate of change of velocity Δv with respect to time Δt:
aav=

Δv

Δt

=

v2−v1

t2−t1

[

m

s2

]
where:
t1 - the time at which the object had velocity v1
t2 - the time at which the object had velocity v2

The instantaneous acceleration can be found by differentiating position with respect to time twice or differentiating velocity with respect to time once:

a(t)=

dv

dt

=

d

dt

(

dx

dt

)=

d2x

dt2

[

m

s2

]

---