Linear Motion

1. Introduction to Linear Motion

Linear motion is one of the most fundamental topics in Engineering Mechanics. It deals with the motion of a body along a straight line. Many engineering systems such as pistons, elevators, vehicles on straight roads, falling bodies, and sliding blocks exhibit linear motion.

Understanding linear motion is essential before studying projectile motion, rotational motion, dynamics, and machine mechanisms.

2. Definition of Linear Motion

A body is said to undergo linear motion when all points of the body move along straight-line paths and have the same displacement, velocity, and acceleration at any instant.

Linear motion is also known as rectilinear motion.

3. Types of Linear Motion

(a) Uniform Linear Motion

• Velocity remains constant
• Acceleration is zero

Example:

• A car moving at constant speed on a straight road

(b) Non-Uniform Linear Motion

• Velocity changes with time
• Acceleration is not zero

Example:

• A freely falling body
• A vehicle accelerating or decelerating

4. Basic Terms Used in Linear Motion

(a) Position

The location of a particle with respect to a fixed reference point.

(b) Displacement

Displacement is the change in position of a body in a particular direction.

• Vector quantity
• Can be positive or negative

$$\text{Displacement} = \text{Final position} – \text{Initial position}$$Displacement=Final position−Initial position

(c) Distance

• Total length of the path travelled
• Scalar quantity
• Always positive

(d) Velocity

Velocity is the rate of change of displacement with respect to time.$$\text{Velocity} = \frac{ds}{dt}$$Velocity=dtds​

• Unit: m/s
• Vector quantity

(e) Speed

• Magnitude of velocity
• Scalar quantity

(f) Acceleration

Acceleration is the rate of change of velocity with respect to time.$$\text{Acceleration} = \frac{dv}{dt}$$Acceleration=dtdv​

• Unit: m/s²
• Can be positive (acceleration) or negative (retardation)

5. Uniformly Accelerated Linear Motion

When acceleration is constant, the motion is called uniformly accelerated motion.

Equations of Motion

For constant acceleration:

$$v = u + at$$v=u+at

$$s = ut + \frac{1}{2}at^2$$s=ut+21​at2

$$v^2 = u^2 + 2as$$v2=u2+2as

Where:

• $u$u = Initial velocity (m/s)
• $v$v = Final velocity (m/s)
• $a$a = Acceleration (m/s²)
• $t$t = Time (s)
• $s$s = Displacement (m)

6. Motion Under Gravity

Motion under gravity is a special case of linear motion where:

• Acceleration = $g = 9.81 \, m/s^2$g=9.81m/s2 (downward)

Free Fall

• Initial velocity $u = 0$u=0
• Acceleration $a = g$a=g

Equations become:$$v = gt$$v=gt $$s = \frac{1}{2}gt^2$$s=21​gt2

7. Retardation (Deceleration)

Retardation is negative acceleration, meaning velocity decreases with time.

Example:

• Application of brakes in a vehicle

$$a = \frac{v – u}{t} \quad (a < 0)$$a=tv−u​(a<0)

9. Relative Linear Motion

Relative motion considers the motion of one body with respect to another moving body.

If:

• Velocity of A = $v_A$vA​
• Velocity of B = $v_B$vB​

Then:$$\text{Relative velocity of A w.r.t B} = v_A – v_B$$Relative velocity of A w.r.t B=vA​−vB​

Used in:

• Train problems
• Vehicle overtaking problems

10. Importance of Linear Motion in Engineering

Linear motion is widely used in:

• Machine design
• Piston-cylinder mechanisms
• Elevators and lifts
• Structural analysis
• Vehicle dynamics

11. Practical Engineering Examples

• Sliding of blocks on inclined planes
• Motion of piston in IC engines
• Vertical motion of cranes
• Conveyor belt systems