Position and Displacement

Position (x): Location of object relative to origin

Displacement (Δx): Change in position

• Formula: Δx = x₂ - x₁
• Vector quantity (has direction)
• Can be positive, negative, or zero

Distance: Total path length traveled

• Scalar quantity (no direction)
• Always positive or zero

Velocity

Average velocity: Displacement over time

• Formula: v_avg = Δx / Δt
• Units: m/s
• Vector quantity

Instantaneous velocity: Velocity at specific instant

• Limit as Δt → 0
• Slope of position-time graph

Speed: Magnitude of velocity (scalar)

Acceleration

Average acceleration: Change in velocity over time

• Formula: a_avg = Δv / Δt
• Units: m/s²
• Vector quantity

Instantaneous acceleration: Acceleration at specific instant

• Slope of velocity-time graph

Kinematic Equations

For constant acceleration:

1. v = v₀ + at
2. x = x₀ + v₀t + ½at²
3. v² = v₀² + 2a(x - x₀)
4. x = x₀ + ½(v₀ + v)t

Where:

• v₀ = initial velocity
• v = final velocity
• a = acceleration
• t = time
• x = position

Exercises - Part 1

• What is displacement?
• ans: Change in position, Δx = x₂ - x₁, vector quantity
• What is the difference between distance and displacement?
• ans: Distance is total path length (scalar), displacement is change in position (vector)
• If you walk 5m forward then 3m backward, what is your displacement?
• ans: 2m forward (5m - 3m = 2m)
• What is your total distance traveled?
• ans: 8m (5m + 3m = 8m)

Exercises - Part 2

• What is average velocity?
• ans: Displacement divided by time, v_avg = Δx / Δt
• A car travels 100m in 5s. What is its average velocity?
• ans: v_avg = 100m / 5s = 20 m/s
• What does the slope of a position-time graph represent?
• ans: Instantaneous velocity
• What is acceleration?
• ans: Rate of change of velocity, a = Δv / Δt

Exercises - Part 3

• A car accelerates from 0 to 20 m/s in 4 seconds. What is acceleration?
• ans: a = (20 - 0) / 4 = 5 m/s²
• What does positive acceleration mean?
• ans: Velocity is increasing in the positive direction
• Can an object have negative velocity and positive acceleration?
• ans: Yes, object moving backward but slowing down
• What is the slope of a velocity-time graph?
• ans: Acceleration

Exercises - Part 4

• Which kinematic equation relates v, v₀, a, and t?
• ans: v = v₀ + at
• A ball drops from rest. After 2s, what is its velocity? (g = 10 m/s²)
• ans: v = 0 + (10)(2) = 20 m/s downward
• How far does it fall?
• ans: x = ½at² = ½(10)(2²) = 20m
• A car going 30 m/s brakes with a = -5 m/s². How long to stop?
• ans: 0 = 30 + (-5)t, t = 6 seconds

Exercises - Part 5

• Using v² = v₀² + 2ax, find stopping distance for car above
• ans: 0² = 30² + 2(-5)x, x = 900/10 = 90m
• What does constant velocity mean?
• ans: Acceleration is zero, v = v₀
• If a = 0, which equation simplifies to x = v₀t?
• ans: x = x₀ + v₀t + ½at² becomes x = x₀ + v₀t when a = 0
• Can displacement be zero while distance is not?
• ans: Yes, if you return to starting point (round trip)

Exercises - Part 6

• A train accelerates at 2 m/s² for 10s starting from rest. Final velocity?
• ans: v = 0 + 2(10) = 20 m/s
• How far does it travel?
• ans: x = ½(2)(10²) = 100m
• Average velocity during this motion?
• ans: v_avg = (0 + 20)/2 = 10 m/s, or x/t = 100/10 = 10 m/s
• What is free fall?
• ans: Motion under gravity only, a = g ≈ 9.8 m/s² downward