Learning Objectives
- Understand the concepts of position, displacement, and distance
- Define and differentiate between speed, velocity, and acceleration
- Derive and apply equations of motion for uniformly accelerated motion
- Analyze position-time, velocity-time, and acceleration-time graphs
- Understand relative velocity in one dimension
Key Concepts
Position, Path Length and Displacement
Position: Location of an object with respect to a chosen reference point (origin).
Path Length (Distance): Total length of the path traversed. It is a scalar quantity and always positive.
Displacement: Change in position vector. It is a vector quantity. Displacement ≤ Distance. Δx = x₂ - x₁.
Speed and Velocity
Average Speed: Total distance / Total time. It is always positive.
Average Velocity: v_avg = Δx/Δt = (x₂ - x₁)/(t₂ - t₁). It can be positive, negative, or zero.
Instantaneous Velocity: v = dx/dt (derivative of position with respect to time).
Instantaneous Speed: |v| = |dx/dt| (magnitude of instantaneous velocity).
Acceleration
Average Acceleration: a_avg = Δv/Δt = (v₂ - v₁)/(t₂ - t₁)
Instantaneous Acceleration: a = dv/dt = d²x/dt²
When velocity and acceleration are in the same direction, the object speeds up. When in opposite directions, it slows down (decelerates).
Equations of Motion (Uniform Acceleration)
For motion with constant acceleration a, initial velocity u, final velocity v, displacement s, and time t:
- First Equation: v = u + at
- Second Equation: s = ut + ½at²
- Third Equation: v² = u² + 2as
- Displacement in nth second: sₙ = u + a(2n - 1)/2
Graphical Analysis
- x-t graph: Slope = velocity. Straight line means uniform velocity. Parabola means uniform acceleration.
- v-t graph: Slope = acceleration. Area under curve = displacement.
- a-t graph: Area under curve = change in velocity.
Free Fall
An object falling under gravity alone (neglecting air resistance) experiences uniform acceleration g = 9.8 m/s² downward.
Equations become: v = u + gt, h = ut + ½gt², v² = u² + 2gh (taking downward as positive).
For an object dropped from rest: h = ½gt², v = gt, v² = 2gh.
Relative Velocity
Velocity of object A relative to B: v_AB = v_A - v_B
If two objects move in the same direction, relative velocity = difference of velocities.
If in opposite directions, relative velocity = sum of velocities.
Summary
Motion in a straight line (rectilinear motion) involves position, displacement, velocity, and acceleration as key quantities. The three equations of motion apply for uniform acceleration. Graphical analysis provides visual understanding of motion parameters. Free fall is a special case with acceleration g = 9.8 m/s². Relative velocity helps analyze motion of one object with respect to another.
Important Terms
- Displacement: Shortest distance between initial and final positions with direction
- Uniform Velocity: Equal displacement in equal time intervals
- Non-uniform Acceleration: Acceleration that changes with time
- Free Fall: Motion under gravity alone without air resistance
- Relative Velocity: Velocity of one object as observed from another moving object
Quick Revision
- Displacement is a vector; Distance is a scalar; |Displacement| ≤ Distance
- v = u + at, s = ut + ½at², v² = u² + 2as
- Slope of x-t graph = velocity; Slope of v-t graph = acceleration
- Area under v-t graph = displacement
- Free fall: a = g = 9.8 m/s² (downward)
- Relative velocity: v_AB = v_A - v_B